







A line cuts a curve 
20181111 

From roxanne: Hello, i need to ask a question, do you mind explaining and writing the formula of how to solve equations such as "find the set values of k for which the line y2x5 cuts the curve y=x^2 +kx+11" please Answered by Penny Nom. 





Three consecutive whole numbers 
20180622 

From whaha: the sum of three consecutive whole numbers divided by 4 is equal to 6. find the numbers. Answered by Penny Nom. 





Two odd integers 
20180604 

From Pleroma: The product of two alternate odd integers exceeds three times the smaller number by 12.
What is the larger number? Answered by Penny Nom. 





Three consecutive odd integers 
20171010 

From Johanna: Three consecutive odd integers are such that the sum of the first and second is 31 less than 3 times the third Answered by Penny Nom. 





Four multiples of 10 
20170923 

From Laudacir: Four multiples of 10 are added together.the total is a 3 digit number with three consecutive digits. What could the four number be? Answered by Penny Nom. 





3 consecutive multiples of 11 
20170722 

From nisha: using the multiples formula shown at ypur site how can we solve finding 3 consecutive multiples of 11 whose sum is 363 Answered by Penny Nom. 





Two consecutive odd integers 
20170714 

From Adeeb: Find two consecutive odd positive integers, sum of whose square is 290. Answered by Penny Nom. 





Three consecutive natural numbers 
20170611 

From Anita: Three consecutive natural numbers are such that the square of the middle number exceeds the difference of the square of the other two by 60. Find the numbers. Answered by Penny Nom. 





Five consecutive even integers 
20160812 

From Marcelle: The average of 5 even consecutive numbers is 112. What is the largest numbers? Answered by Penny Nom. 





Consecutive angles of a parallelogram 
20160128 

From Hanna: The consecutive angles of a parallelogram measures Answered by Penny Nom. 





Three consecutive integers 
20160107 

From ronn: Three consecutive integers whose sum is 186 Answered by Penny Nom. 





Four consecutive integers 
20160105 

From sherica: my teacher gave us an assignment and it says here that;
THERE ARE FOUR CONSECUTIVE INTEGERS.IF THE SMALLEST ONE IS x ,FIND THE SUM OF THE FOUR INTEGERS. that is the question ...thanks hope that u can answer it Answered by Penny Nom. 





The surface area of a cut pipe 
20150427 

From ARUN: dear Sir,
please advise me , how to calculate the surface area of a pipe with diameter of 630 mm and thickness of 67 mm which is cut in a angle of 22.5 degree.
please show me how to calculate the surface area of the pipe which cut in an angle.
Thanking you. Answered by Harley Weston. 





Three consecutive integers 
20150328 

From Aleksey: I need help how do I find the second of three consecutive integers if the sum of the first and third is 26? Answered by Penny Nom. 





Two consecutive odd integers 
20150309 

From gowtham: The sum of two consecutive odd integers is 20.find the integers Answered by Penny Nom. 





Two consecutive positive integers 
20150301 

From meha: Find two consecutive positive integers , sum of whose squares is 365. Answered by Penny Nom. 





Three consecutive odd integers 
20141017 

From Maddy: 3 times the largest of three consecutive odd integers is 13 more than the sum of the other two. What are the integers? Answered by Penny Nom. 





Cutting a round cake so that it doesn't dry out 
20140826 

From James: I'm wondering if there's a simple way to calculate the area between two parallel chords of a circle equidistant from its diameter, or if I have the area, to find the distance between the two chords.
Here's my "problem". You may have heard of the way of cutting a round cake so that it doesn't dry out  make two parallel cuts (chords) the length of the cake, take the middle piece, then push the two pieces together.
So I know the area of a 12" cake, and I want say, exactly an eighth of the cake. How wide do I cut the centre piece?
Now to get even more difficult, the next day I want another eighth from the centre. How wide do I cut the next pieces, and so on...?
Thanks,
James Answered by Harley Weston. 





A word problem with fractions 
20140326 

From lois: a man has a stack of boards, he cuts 3/5 off each board, after he cuts the boards he finds he has enough pieces left to make 4 boards the length of the original boards. how many boards did he start with. Answered by Penny Nom. 





Three consecutive whole numbers 
20140313 

From Abhishek: Three consecutive whole numbers are such that if they be divided by 5,3 and 4 respectively, the sum of the quotient is 40. What are the numbers? Answered by Chris Fisher. 





Consecutive Numbers Question 
20130811 

From Lucy: Hello,
I have a question about consecutive numbers. So lets say a question tells you to find what three consecutive numbers are that add up to a certain number. To calculate this, would you use
2x+1, 2x+3, 2x+5 or just x, x+2, x+4.
Also, the same question for even numbers. Would you use:
2x, 2x+2, 2x+4 or just x, x+2, x+4.
My math book is confusing because it uses the first strategy(the 2x+…) in some situations and the second strategy(x+…) in other situations. Thus, can you provide me an explanation of why you would use the strategy that is correct?
Specifically, my book answered this question: If “n” is the first of five consecutive odd numbers, what is their average, using n,n+2….=5n+20 and average is n+4(this is the answer),.
For the question: the sum of 4 consecutive odd integers must be:
Divisible by 8 but not necessarily 16(this is the answer), it used the strategy 2k+1, 2k+3…=8k+16.
Thank you so much for taking the time to answer my question. Answered by Harley Weston. 





Three consecutive odd integers 
20130708 

From sparsh: Three times the first of three consecutive odd integers is 3 more than twice the third. The third integer is: Answered by Robert Dawson. 





Three consecutive numbers 
20111129 

From Michael: Find three consecutive numbers such that the second number squared
is equal to the first and third added together Answered by Penny Nom. 





Three consecutive terms of an arithmetic sequence 
20111030 

From Juliette: Find X when x, 1/2x + 7, 3x 1 are consecutive terms of an arithmetic sequence. Answered by Penny Nom. 





Three consecutive odd integers 
20111007 

From Muneera: Find 3 consecutive odd integers so that 3 times their sum is 5 more than 8 times the middle one? Answered by Penny Nom. 





Three consecutive odd numbers 
20110311 

From George: Write an equation three consecutive odd numbers whose sum is 75 Answered by Penny Nom. 





Three consecutive numbers 
20110131 

From parth: name 3 consecutive numbers,each less then a 100,the smallest # is divisible by 6 the next is divisibel
by 5 and the largest divisible by 4 Answered by Penny Nom. 





Three consecutive odd integers 
20101107 

From karen: find three consecutive odd integers such that twice the sum of the second and the third is 43 more that three times the first Answered by Penny Nom. 





Four consecutive integers 
20101021 

From Mickey: Find four consecutive integers such that the sum of the first three integers is 20 more then the fourth integer.
I know that for consecutive integers the algebraic expression is x+x+1+x+2+x+3 Answered by Penny Nom. 





The product of 2 consecutive integers 
20100628 

From Sharon: the product of 2 consecutive integers is 6 less than the square of the larger number.
find the smaller number Answered by Penny Nom. 





The product of two consecutive even integers 
20100511 

From Terry: The product of two consecutive even integers is 14 more than their sum. Find the integers. Answered by Penny Nom. 





Consecutive even integers 
20100203 

From cassandra: find 2 consecutive even integers whose sum is 34
find 3 consecutive even integers such that 3 time the middle interger is 10 more than the greatest. Answered by Tyler Wood. 





A cylinder is cut at an angle of 20 degrees 
20100125 

From sanjay: If I have a cylinder with 40mm diameter and I will cut it at an angle
20 degrees so what would be the increased radius or dia.??? of the cylinder Answered by Penny Nom. 





Improper fractions 
20100112 

From Blanca: Between which two consecutive natural numbers does the improper fraction 57/56 lie? Answered by Robert Dawson. 





Consecutive odd intergers 
20100106 

From keyanna: list five consecutive odd integers, starting with 3 Answered by Robert Dawson. 





Cutting a square into four shapes 
20100105 

From Tracy: A square with a side of 4 ft needs to be cut into four shapes with the same area. can you find six different was to cut the square? Answered by Robert Dawson. 





Consecutive odd integers 
20091109 

From Erin: I'm having trouble with consecutive integers. i know it goes n + n + 1 + n + 2
and so on, but if its odd then do the numbers change to odd ones? i don't know
the question is: the sum of four consecutive odd integers is 72. write an equation to model
this situation and find the values of the four integers Answered by Robert Dawson and Penny Nom. 





sinA + cosA = 1 
20091106 

From Nazrul: What is the solution of sinA+cosA=1 where A is an acute angle? Will A=0 degree be included in the solution? Please explain. Answered by Harley Weston. 





Consecutive positive integers 
20091101 

From Madi: I'm doing my homework, and there are questions on Consecutive Positive Integers. I was just wondering, what is a consecutive integer?
Or to be more specific, what is a consecutive POSITIVE integer? Answered by Penny Nom. 





Cutting a pipe at an arbitrary angle 
20091024 

From Carol: I have read your article on cutting pipe,etc. at a 45 degree angle. I need to develop
an equation and pattern for cutting any size pipe (3" to 7") to any degree. I don't
understand how to transfer the wave pattern to graph paper.
Thanks. Answered by Harley Weston. 





Four consecutive odd integers 
20091024 

From Tonya: the sum of four consecutive odd intergers 2008 what are the intergers Answered by Penny Nom. 





Cubes and squares 
20090916 

From Stanley: What is a three consecutive digit number like 5,6,7 , which is two less than a cube and two more than a square? Answered by Robert Dawson. 





Is 100 degree angle an acute angle? 
20090902 

From Nazrul: Is 100 degree angle an acute angle? Please explain. Here 100<90. Answered by Robert Dawson. 





Zero Angle 
20090901 

From Nazrul: Is zero degree an acute angle? Please explain. Answered by Janice Cotcher. 





Three consecutive odd integers 
20090402 

From Brian: Find the greatest of three consecutive odd integers if the product of the second and third is 8 more than 13 times the first. Answered by Robert Dawson. 





Three consecutive odd integers 
20090311 

From Susan: Find three consecutive odd integers such that the sum of the first, twice the second, and three times the third is 70 Answered by Penny Nom. 





The angles of a triangle 
20090311 

From Marissa: The angles in a triangle measure 7x1, 18x+2, and 5x+10. Determine whether the
triangle is acute, obtuse, or right. State your reasons clearly. Answered by Robert Dawson. 





wo consecutive numbers with a product of 9506 
20090225 

From sofie: find two consecutive numbers with a product of 9506 Answered by Penny Nom. 





A railway cut 
20090130 

From Jean: A railway cut 200ft. long and 30ft. wide at the bottom is made with sloping sides, which are 80ft. and 60ft. in length. The 80ft. side is inclined 45 degrees and the other side is 35 degrees to the horizontal. Find the cost of removing the earth at $2 per load, if the trucks have capacity of 4cu. yd. Answered by Penny Nom. 





Three consecutive integers 
20081007 

From pam: Find three consecutive integers such that the greatest increased by twice the least is 293. Answered by Penny Nom & Chris Fisher. 





Two consecutive numbers 
20080921 

From tomas: what is the two consecutive numbers with a product of 4160? Answered by Penny Nom. 





Cutting a pipe on a 45 degree angle 
20080912 

From Bakshani: how do you a mark a pipe 5 inch diameter and cut it to form a 45 degree angel Answered by Stephen La Rocque. 





Three consecutive positive integers 
20080720 

From AC: find three consecutive positive integers such that the product of the first and third, minus the second, is 1 more than 4 times the third Answered by Penny Nom. 





Product of Two Integers 
20080625 

From devin: two consecutive integer have a product of 240. what are they Answered by Penny Nom. 





Two consecutive negative even numbers 
20080603 

From Chelsea: The sum of the squares of two consecutive negative even numbers is 100. Find the integers, Answered by Penny Nom. 





Two consecutive odd integers 
20080324 

From Jarrod: the sum of two consecutive odd integers is 24. Find the two integers Answered by Stephen La Rocque. 





Two consecutive odd positive integers 
20080312 

From Sarah: The sum of the squares of two consecutive odd positive integers is 74. What are th integers? Answered by Penny Nom. 





3 consecutive odd integers 
20080228 

From Pat: For 3 consecutive odd integers, twice the first integer, plus 5 more than the second integer, plus three less than four times the third integer equals 85. Find the 3 consecutive odd integers. Answered by Penny Nom. 





Classifying a triangle 
20080207 

From kevin: scalene triangle 8 ft base right side 9.5 left side 12 ft what is the angles Answered by Penny Nom. 





Three consecutive odd integers 
20080124 

From daniel: find three consecutive odd integers such that three times the second minus the third is 11 more than the first.
help! i have x+1, x+3, x+5 as my 3 odd integers is this correct its been a while. Answered by Penny Nom. 





Find the lengths, and explain in a 'let statement' 
20071220 

From Cordilla: the length of the sides of a triangle are 3 consecutive whole numbers. The perimeter of the triangle is 102 meters. Find the lengths, and explain in a' let statement'. Answered by Stephen La Rocque and Harley Weston. 





Three consecutive numbers 
20071208 

From Aris: The product of 3 consecutive numbers is 1716. Find their sum. Answered by Stephen La Rocque. 





the sum of two consecutive integers is 51 
20071017 

From Keira: the sum of two consecutive integers is 51 Answered by Penny Nom. 





Four consecutive integers 
20070928 

From Joshua: We were given a question that involved consecutive integers. I can not seem to solve this question and I am the top student in my class.The question states: Find four consecutive integers such that twice the sum of the two greater integers exceeds three times the first by 91. How do you solve this question? Thank you very much for your support. Answered by Penny Nom, Stephen la Rocque and Claude Tardif. 





The sum of two or more consecutive integers 
20070426 

From nana: some numbers can be written as the sum of two or more consecutive integers(we consider positive integers only)
other number (eg. 4) cannot be express in this way.
let us call a number which can be expressed in this way a SOALTCI( sum of at least two consecutive integers)
a). list the first few SOALTCI and conjecture the general formula.
b). Prove that any number of the given form(in your answer (a)) is a SOALTCI Answered by Penny Nom. 





Determine the probability that the triangle is acute. 
20070410 

From Greg: Points A1, A2, . . ., AN are equally spaced around the circumference of a
circle and N >=3. Three of these points are selected at random and a triangle
is formed using these points as its vertices. If N = 2k for some positive integer
k >= 2, determine the probability that the triangle is acute. Answered by Penny Nom. 





Two consecutive odd numbers 
20070319 

From Alicia: The sum of two consecutive odd numbers is divisible by 4.
Can you make similar statements about the sum of three consecutive odd
numbers and of four consecutive odd numbers?
Generalize your findings.
I tried :
n= smallest odd number
therefore the next = n, n+2 and the second =n+4 and the third = n+6
Now n+ n+2+n+4+n+6/4
Please help I am not sure of this I am just trying Answered by Penny Nom. 





Choosing numbers with no two consecutive 
20070315 

From Meghan: How many combinations are there of 3 numbers chosen from 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 but where no 2 of the 3 numbers are consecutive?
Thank you!
Meghan Answered by Penny Nom. 





Can the triangle be called both an acute triangle and an equilateral triangle? 
20070310 

From Jane: If all sides of a triangle are the same length and all angles are 60 degrees, can the triangle be called both an acute triangle and an equilateral triangle? Answered by Walter Whiteley. 





Three consecutive odd integers 
20070108 

From Emma: Find three consecutive odd integers such that the sum of the first and second is 27 less than 3 times the third. Answered by Stephen La Rocque. 





Twice the smaller of two consecutive integers is the larger increased by 5. 
20070108 

From Evan: Twice the smaller of two consecutive integers is the larger increased by 5. find the integers. Answered by Stephen La Rocque and Penny Nom. 





The sum of 2000 consecutive integers is 1000 
20061029 

From Matias: if the sum of 2000 consecutive integers is 1000, then the sum digits of the greatest of these 2000 integers is? Answered by Penny Nom. 





Three consecutive integers 
20061029 

From Jimmy: Basically i want to know how you prove that the product of any 3 consecutive integers is a multiple of 6 Answered by Stephen La Rocque. 





Three consecutive odd numbers 
20061021 

From Pige: The sum of three consecutive odd numbers is 387. Find the numbers. Answered by Penny Nom. 





A board is cut into two pieces... 
20060711 

From DeAnna: A 102" board is cut into 2 pieces. One piece is two times the length of the other. Find the lengths of the two pieces. Answered by Stephen La Rocque. 





We are 2 pairs of consecutive 2 digit numbers ... 
20060513 

From A perent: We are 2 pairs of consecutive 2 digit numbers that are factors of the number of feet in a mile. Who are we? Answered by Claude Tardif. 





Adding consecutive numbers 
20060426 

From Lisa: When I have a total that is the sum of consecutive numbers, how do I figure out what the numbers are?
Answered by Stephen La Rocque. 





Divide the product of three consecutive positive integers by their mean 
20060312 

From Sandy: When I divided the product of three consecutive positive integers by their mean, I got 99. What is the smallest of the three numbers? Answered by Stephen La Rocque. 





Two consecutive even integers 
20060311 

From Cheryl: Find two consecutive even integers that have a sum of 450
Answered by Stephen La Rocque. 





Find two consecutive integers whose product is 182 
20060209 

From Anthony: Find two consecutive integers whose product is 182 Answered by Steve La Rocque. 





The sum of n consecutive integers 
20051127 

From Craig:
If 875 is expressed as the sum of n consecutive integers (n > 1), then the number
of possible values of n is?
Possible Answers:
A) 15
B) 8
C) 7
D) 16
E) 4
Answered by Penny Nom. 





Adding consecutive integers 
20050513 

From Yas: i need some help with my work which is to investigating sums of consecutive numbers for example: 15= 4+5+6 and 15=1+2+3+4+5 Answered by Penny Nom. 





The least integer of a set of consecutive integers 
20050419 

From Lucia: The least integer of a set of consecutive integers is 25. If the sum of these integers is 26, how many integers are in the set? Answered by Penny Nom. 





The length of a cut 
20040917 

From Florita: My daughhter, who is a 9th grader is attempting to cut a piece of wood after determining the length of the cut for the hypotenuse. These are the measures:
a=4squared, b=6squared.
She determined that c should equal 52. But when she measured the actual piece to be cut, c measured 39.5 inches! Can you offer any insight as to what she is doing wrong? I have suggested that she may be working with an Acute rather than a Right angle . But she insists that it is a Right angle after using a "framing square". Answered by Claude Tardif. 





The product of any n consecutive integers 
20040118 

From Fazia: I have to prove that the product of any n consecutive integers is divisible by n!(eg:the product of five consecutive integers is divisible by 5!) It's easy enough to plug in numbers and prove it is true, but i'm finding it difficult to come up with a proper proof. Answered by Claude Tardif. 





Two consecutive negative integers 
20040117 

From Amanda: Find two consecutive negative intergers whose product is 182. Answered by Penny Nom. 





Consecutive integers 
20030907 

From Amy: I was wondering if there was a proper way to write the solution to a problem requiring consecutive integers? Is it okay to write the solution as 5,4,3,2,1 or must it be written as 1,2,3,4,5? Answered by Penny Nom. 





39 consecutive natural numbers 
20030819 

From A student: Prove that among any 39 consecutive natural numbers it is always possible to find one whose sum of digits is divisible by 11. Answered by Penny Nom. 





Cutting some wood 
20030725 

From Betty: My husband is building a six sided wood circle and would like to know the angle to cut the pieces. Answered by Penny Nom. 





A pyramid with its top cut off 
20030721 

From David: What is the name given to a 3D shape that looks like a pyramid with its top cut off? Answered by Penny Nom. 





positive multiples of 10 that are the sum of four consecutive integers 
20030501 

From Taurus: How many positive multiples of 10 that are less than 1000 are the sum of four consecutive integers? Answered by Penny Nom. 





The product of four consecutive integers is 3024. 
20030425 

From pillar: The product of four consecutive integers is 3024. What are those numbers. Answered by Penny Nom. 





Divide a circle in 8 equal pieces 
20030404 

From Naomi: I have to divide a circle in 8 equal pieces but can only cut 3 times can you please help me Answered by Penny Nom. 





Three consecutive positive intergers 
20030209 

From Yew: Prove that when we multiply any consecutive positive intergers, the result is always divisible by 6.
ex. (7)(8)(9) = 504 = 6 (84) Answered by Penny Nom. 





Cutting a board 
20030120 

From George: A person has a sheet of board which he saws into two (2) pieces. The length of the first piece is two thirds the length of the original board, while the length of the second piece is four feet longer than the first piece. How long was the original board? (the length is defined as the longer of the two sides of the board) Answered by Penny Nom. 





Express 5120 as a sum of consecutive numbers 
20020825 

From Todd: Express 5120 as a sum of more than 1 consecutive number. Answered by Leeanne Boehm. 





Third side of Acute Triangle 
20011109 

From Don: I gave this problem to my children, 9th and 12th grade and they are not(me too!) sure of the answer. I am trying to determine the length of the third side of an acute triangle. I know two of the lengths but I do not know the angles. Is there a proper formula to use to find the length of the third side? Thank you. Answered by Chris Fisher and Penny Nom. 





An acute scalene triangle 
20010924 

From Kim: Hi I'm on a missionI'm suppose to have a group of children write up a hypotheses about the altitudes and angle biscetors of the follwing 3 types of triangleequilateral, isosceles and acute scalene triangle. Is there such a thing as a acute scalene triangle and why or why not. Answered by Chris Fisher. 





Sharing a donut 
20010906 

From Amanda: You have invited 11 people over to your house one day, and your friends are hungry. You go into the kitchen and find out you only have 1 donut (with a hole in the middle). In order to feed 12 people (including you), you must cut the donut into 12 pieces with only using a straight knife and cutting 3 times. This is NOT a trick question. Answered by Claude Tardif. 





4160 
20010307 

From Jessica: The product of two consecutive page numbers in this book is 4160. What are the page numbers? Answered by Penny Nom. 





A pole next to a building 
20010220 

From Robert L. Haun: A pole 100 feet tall is erected next to a building 10' X 10' X 10' . Object is to cut off the pole so the upper end remains touching the part cut off, while the cutoff portion touches the cutoff point, the edge of the building and the ground. Note assume the pole is touching the building and is "very thin" What is/are the cutoff points? Answered by Penny Nom. 





Crown molding mitre cuts 
20001106 

From Jim Tomfohrde: My question has to do with making mitre cuts when installing crown molding. Crown molding is the trim that is put up at the top of walls with one edge on the wall and the other edge on the ceiling. To make a mitre cut on your mitre saw for a 90degree corner you can lay the molding flat on the saw base, set the bevel of the blade to 34 degrees and the mitre to 31.5 degrees (these may be slightly appoximate). Of course depending on which piece of molding you're cutting you will cut one end or the other, or use the left or right end. These angles allow the cuts to line up and form a seamless corner when they're put in place on the wall/ceiling at 90 degrees. My question is this  is there some mathematical formula from which the 34 degrees and 31.5 degrees are derived. I want to know this because in many cases the corner is not 90 degrees but can be more or less, and in these instances I would like to know if I can calculate the bevel and mitre to use based on the angle of the corner. Answered by Harley Weston. 





Cutting the lawn 
20001026 

From Kacie: Ellen can mow a lawn in 2 hours. Mary can mow the same lawn in 1.5 hours. How long would it take them to mow the lawn together? Answered by Penny Nom. 





Three consecutive odd integers 
20000818 

From Wallace: A sixdigit integer XYXYXY, where X and Y are digits is equal to five times the product of three consecutive odd integers. Determine these three odd integers. Answered by Paul Betts. 





Cutting a carpet 
19991215 

From Heather: A rectanglular piece of carpeting is 90 inches long and 90 inches wide. How can the carpet be cut into two pieces of equal sides and shape to cover an area of 100inches? There can only be one cut and no scraps. Please show me how. Answered by Penny Nom. 





Ratios and proportions 
19991119 

From Rebecca: I'm not that good of a student and I was woundering if there is a shortcut to ratios and proportion. My 6th grade teach tought all we needed to know. I was just wounder if there was an easier way. Answered by Claude Tardif and Harley Weston. 





Cutting a board into 8 equal pieces 
19991021 

From Brittany: Jamie wanted a board cut into 8 equal pieces. The Lucas Lumber Co. charges 60 cents for cutting a board into 4 equal pieces. How much will it charge for cutting Jamie's board? Answered by Claude Tardif. 





Deux nombres consécutifs 
20100118 

From Janot: Bonjour
comment peut on faire pour démontrer
que si deux nombres sont consécutifs, leur produit et leur somme sont
premiers entre eux
Merci de me mettre sur la voie , j'ai échoué avec Bezout. Answered by Claude Tardif. 





La somme de 51 nombres entiers consécutifs est 1785 
20030401 

From Katrine et Armand: La somme de 51 nombres entiers consécutifs est 1785, quels sont ces nombres? (indication : on rappelle que pour tout entier P, on a 1 + 2 + ....+ P = P (P + 1) / 2). Answered by Claude Tardif et Penny Nom. 

