







Loan payment formula 
20190224 

From Kenneth:
I have a question regarding the loan payment formula shown below.
Calculating the Payment Amount per Period
The formula for calculating the payment amount is shown below.
Simple Amortization Calculation Formula
A = P X r(1 + r)n over (1 + r)n  1
where
A = payment Amount per period
P = initial Principal (loan amount)
r = interest rate per period
n = total number of payments or periods
Is this formula/calculation a condensed version of a longer calculation? I am curious to know how the (1 +r)n  1 was developed from the longer calculation. For example, r(1 + r)n may have been (r + rn)n. The n's are exponents.
I thank you for whatever helpful explanation that may be provided.
Kenneth Answered by Harley Weston. 





A geometric series 
20180313 

From nathi: Hi I am really struggling with this question please help !!!!
a pohutukawa tree is 86 centimetres when it is planted. in the first year after it is planted , the tree grows 42 centimetres in height.Each year the tree grows in height by 95% of the growth of the previous year.
assume that the growth in height of the pohutukawa tree can be modelled by a geometric sequence.
A)find the height of the tree 5 years after it is planted and figure out the maximum height the pohutukawa tree is expected to reach in centimetres.
The maximum height part is not answered. Answered by Penny Nom. 





A road trip word problem 
20160410 

From Marc: While on a road trip I imagined this random word problem. Suppose I start a trip of 75 miles. My initial speed is 75 miles per hour. After every mile traveled I decrease my speed by one mile per hour. After the first mile I decrease my speed to 74 miles per hour and so on for each subsequent mile traveled. How long will it take to complete the 75 mile journey? Answered by Penny Nom. 





Which term of the series 2+7+12+...is 152? 
20160201 

From francis: whice term of the series 2+7+12+...is 152? Answered by Penny Nom. 





The sum of the first 50 terms of an arithmetic progression 
20140726 

From Joshua: Hello ...my is Joshua...I'm a grade 11 student...I got a question
Calculate the sum of the first 50 terms of an arithmetic progression: 112:98:84 Answered by Penny Nom. 





An infinite geometric series 
20131224 

From Muhammad: The sum of an infinite geometric series is 15 and the sum of their squares is 45. Find the series Answered by Penny Nom. 





A sequence or a series? 
20130905 

From Rahul: Whether the following sequence is increasing or decreasing? I even do not kow whether to call it a sequence or not. an = (1/(1+n))+(1/(2+n))+....+(1/(n+n)).
I am confused. It looks like a series to me. Please help.
Regards,
Rahul Answered by Robert Dawson. 





An arithmetic progression 
20120822 

From A student: the 3rd term of an A.PPP is 10 more than the first term while the 5th term is 15 more than the second.find the sum of the 8th and 15th terms if the 7th term is 7 times the first term. Answered by Penny Nom. 





1+2+4+8....= 1 
20120402 

From Andy: In this minutephysics video, it's claimed that 1+2+4+8....= 1
Is this true, and if so, how?
< href="http://www.youtube.com/watch?v=kIq5CZlg8Rg">http://www.youtube.com/watch?v=kIq5CZlg8Rg Answered by Robert Dawson. 





A tree growth modelled by a geometric series 
20120208 

From Steph: Hi I am really struggling with this question please help !!!!
a pohutukawa tree is 86 centimetres when it is planted. in the first year after it is planted , the tree grows 42 centimetres in height.Each year the tree grows in height by 95% of the growth of the previous year.
assume that the growth in height of the pohutukawa tree can be modelled by a geometric sequence.
A)find the height of the tree 5 years after it is planted and figure out the maximum height the pohutukawa tree is expected to reach in centimetres Answered by Penny Nom. 





1 + 3 + 3^2 ...+3^(n1) = 3^n  1/2 
20120127 

From Vicki: I am trying to find out how to do show how this proof was worked.
Here is the end result 1 + 3 + 3^2 ...+3^(n1) = 3^n  1/2
This equation was used to find the number of white triangles in the Sierpinski Triangle Answered by Walter Whiteley. 





The sum of a series 
20111107 

From Rattanjeet: Find the sum of 1(1/2) + 2(1/4) + 3(1/6) + 4(1/6)(3/4) + 5(1/6)(3/4)2 + 6(1/6)(3/4)3+ ... where 1/6 + (1/6)(3/4) + (1/6)(3/4)2 + ... constitutes a geometric series. Answered by Penny Nom. 





Infinite Logarithmic Series 
20110808 

From Sourik: Dear Expert,
In my Amithabha Mitra and Shambhunath Ganguly's "A Text Book of Mathematics" I found the formula of log (1+x) where the base is e and x lies in between 1 and +1.As I want to learn Mathematics,I am not satisfied with the mere statement of the formula.Please help giving me the full proof.
Thanking you,
Sourik Answered by Robert Dawson. 





An imaginary infinite geometric tree 
20110218 

From Elise: An imaginary infinite geometric tree grows 1m the first day.
2nd day 2 branches and right angles to each other and each 0.5 m long.
3rd day two new branches at ends of each of previous days' 2 branches, again at right angles, and only .25m long each.
And so on, infinitely.
Q: Use relationships of rightangled triangles and high school level knowledge of geometric series to show
the tree height is limited to (4 + sqrt2)/3 m and width to (2(sqrt2 + 1))/3 m. Answered by Robert Dawson. 





A Taylor polynomial for (lnx)/x 
20100929 

From Dave: I have a series problem that I cannot solve. The problem asks for you to compute a Taylor polynomial Tn(x) for f(x) = (lnx)/x. I calculated this poly out to T5(x) and attempted to use this to identify a pattern and create a series in order to calculate Tn(x). However, the coefficients on the numerator out to F5prime(x) are as follows: 1, 3, 11, 50, 274... Ok, so the negative is an easy fix > (1)^n1. But the other coefficients are stumping me. I can't see any sort of pattern there and I've tried every trick I know. Is there another way to go about this?
Thanks! Answered by Chris Fisher. 





A geometric progression 
20100430 

From Kalyani: sum of infinite geometric progression is 9 and common ratio is 1/10
then sum up to 8 terms is? Answered by Chris Fisher. 





1/1+1/2+1/3... 
20100331 

From Mohd: you explained the way of getting the summation of 100 series numbers
but what is the summation of the numbers 1/1+1/2+1/3+1/4+1/5.............................+1/100 Answered by Robert Dawson. 





A sequence of letters and numbers 
20091110 

From Maria: What is the last ten letters and numbers in the following series and how do I work it out so I can explain it to an eleven year old.
J1F8M1A0M1J0J1 Answered by Robert Dawson, Claude Tardif and Harley Weston. 





An infinite geometric series 
20090518 

From terri: find the sum of the infinite geometric series
14 7 +7/2 7/4 +.....
A. 7007/13 B. 2002 C. 28/3 D.5005/7 Answered by Stephen La Rocque. 





2+6+12+20+30+42 
20090424 

From fredy: what is the sigma notation for the series 2+6+12+20+30+42? Answered by Stephen La Rocque. 





The ratio test 
20090306 

From Haley: Use the ratio test to determine whether the series in convergent or divergent.
(2/5)+(4/10)+(8/15)+... Answered by Harley Weston. 





Convergent or divergent? 
20090303 

From Betsy: Use the ratio test to determine if the series is convergent or divergent.
1+1/2^2+1/3^3+1/4^4
I know the general terms are a(n)=1/n^n and a(n+1)=1/(n+1)^(n+1)
but I can't simplify r=lim(n>inf.)=n^n/(n+1)^(n+1) Answered by Robert Dawson. 





The middle term of an arithmetic sequence 
20081215 

From Leigh: Find the sum of the first fifteen terms of an arithmetic series if the middle term is 92 Answered by Penny Nom. 





An arithmetic series 
20081017 

From Laura: In an arithmetic series 5+9+13+...+tn has a sum of 945. How many terms does the series have?
What formula do I use? Answered by Penny Nom. 





Demographics 
20080725 

From shahrukh: Each year for 10 years ,the population of a city increased by 5% of its value in the previous year.
If the initial population was 200 000 ,what was the population after 10 years ?? Answered by Penny Nom. 





A series solution of y' = xy 
20080703 

From sasha: I've to find the power series solution of the differential equation: y' = xy.
I don't know how to find the recursive equation. Can you please help me. Thanks Answered by Harley Weston. 





What number best completes the series? 
20071112 

From Grace: What number best completes the series?
2 3 7 13 27 ____ Answered by Stephen La Rocque, Penny Nom and Harley Weston. 





Applications of sequences and series 
20070827 

From Trish: I'm a grade 12 learner working on a math project based on sequences and series. I'd like to know the different types of sequences and series such as fibonacci, fourier, farey, etc.
I've already used the Fibonacci Sequence and Harmonic Series
and need two more.
The simpler the sequence or series type the better.
I'd also like to know in which nonmathematical areas use sequences and series
and how.
Areas such as engineering or science. Answered by Penny Nom. 





A sequence of circles 
20070611 

From Ann: Please help with solving the following problem!!!
A circle is inscribed in an equilateral triangle with a side of length 2.
Three circles are drawn externally tangent to this circle and internally
tangent to 2 sides of the triangle. 3 more circles are drawn externally
tantgent to these circles and internally tangent to 2 sides of the triangle. if
this process continued forever, what would be the sum of the areas of all the
circle? the answer 1 parent came up with was Pie over 2, but we don't
know how he did it. Can you please show the work or explain the answer to
this problem?
Thank you
Ann
p s my daughter is in 9th grade math. Answered by Steve La Rocque, Chris Fisher and Penny Nom. 





A geometric series 
20070403 

From jessica: If a geometric series includes 5418+62 as its fifteenth through eighteenth terms, find the sum of the second through the fifth term, inclusive. Answered by Stephen La Rocque. 





Arithmetic Series 
20070218 

From Krista: Question The sum of the first 4 terms of an arithmetic series is 8 and the sum of the first 5 terms is 500. Determine the sum of the 3 terms. Answered by Stephen La Rocque. 





An arithmetic series 
20061128 

From Jillian: Find the sum of 21 terms of an arithmetic series that has an eleventh term equal to 30. Answered by Penny Nom. 





Sigma from 0 to infinity of (n^3 / 3^n) 
20061115 

From Cedric: I'm wondering how you would find if this series converges or diverges?
Sigma from 0 to infinity of (n^3 / 3^n)
Does the n^3 dominate, or does the 3^n dominate? What about higher powers like n^10 / 10 ^ n ? Which one would dominate then? Answered by Penny Nom. 





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. 





Arithmetic Sums 
20060412 

From Angel: (a) In a particular arithmetic sequence, u6 = 344.5 and u20 = 88.3. Find S28.
(b) In a particular arithmetic series, S10 = 495 and S15 = 1005. Express S15 in sigma notation. Answered by Stephen La Rocque. 





Sum of a Geometric Sequence 
20060411 

From Andre: In a particular geometric sequence, U3=3.6 and U10=12.9. Find S16 and S17. Answered by Stephen La Rocque. 





Arithmetic progressions 
20060131 

From A student: 1)the sum to n terms of a particular series is given by S_{n}=17n3n^{2}
a)find an expression for the n term of the series
b)show that the series is an arithmetic progression
2)a particular arithmetic progression has a positive common difference and is such that for any three adjacent terms ,three times the sum of their squares exceeds the square of their sum is 375.Find the common difference
Answered by Penny Nom. 





An Arithmetic sequence 
20051201 

From Aana:
The first term in an arihmetic series is 25 and the 3rd term is 19. Find the number of terms in the series if its is 82.
Here's what I did to find d
a+2d=19; 25+2d= 19 ;1925=2d d=6/2=3
This is where I'm stuck. Can you please provide me with some guidance.
Answered by Penny Nom. 





The sum of a series 
20050831 

From Aamod: Find the sum of the given series till n terms:
(1^{4}\1*3) + (2^{4}\3*5) + (3^{4}\5*7)............... till n terms Answered by Chri Fisher. 





The Maclaurin series generated by f(x)=x^ cosx + 1 
20050810 

From Latto: f(x)=x^{3}·cosx + 1. but when I take the derivatives, I couldn't see a pattern. Can you help?
Answered by Penny Nom. 





A geometric sequence 
20050621 

From A student:
The first three terms of a geometric series are 3(q+5), 3(q+3), (q+7) respectively.
Calulate the value of q. Answered by Penny Nom. 





A Taylor series for ln(x) 
20050416 

From Anood: i have to represent ln(x) as a power series about 2
i`m not getting the final answer which is ln 2+ sigma (((1)^{(n+1)}/
(n*2^{n}))*(x2)^{n}). i don`t get the ln 2 part
i show you my trial
f(x)= ln x.
f(x)=(1/x) .
f(x)= (1/x^{2})*1/2!
f(x)= (2/x^{3})*1/3!
f(x)= (6/x^{4})* 1/4!
so the pattern shows me that f(n)= ((1)^{(n+1)})/x^{n} *n)
so f(2)= sigma ((1)^{(n+1)})/2^{n} *n) *(x2)^{n}
so as you see i don`t get ln 2
Answered by Penny Nom. 





Sum to n terms of the series i.(2^(ni)) 
20050128 

From Satya: Sum to n terms of the series i.(2^{ni}) Answered by Penny Nom. 





The third derivative 
20041015 

From Holly: Why would you ever take the third derivative? Answered by Harley Weston. 





The series from i=1 to n :ai 
20040818 

From Ken: a.) Explain the difference between
the series from i=1 to n :ai
and the series j=1 to n :aj
b.) Explain the difference between
the series from i=1 to n:ai
and the series from i=1 to n :aj Answered by Penny Nom. 





Programming without trig functions 
20040525 

From Derek: I am a programmer trying to calculate the following.
What is the formula to find the crosssectional area of a cylinder with out using any trig functions? or better yet, how can you calculate any given volume in a cylindrical tank with spherical heads with out trig functions?
I am using a PLC (programmable logic controller) to do this and trig functions are not available. Answered by Harley Weston. 





A geometric sequence 
20040413 

From Michael: In a geometric series, the sum of the 2nd and 3rd terms is 60, and the sum of the 3rd and 4th terms is 240. Find the sum of the first 7 terms. Answered by Penny Nom. 





Cosine of 35 degrees 
20040303 

From Jason: How do you find the exact solution to cosine 35 degrees. Answered by Chris Fisher. 





A worm crawling home 
20040218 

From Cindy: A worm is crawling to his home which is one meter away. The longer he crawls the weaker he gets and the less he can crawl the next day. If he crawls within 1/3000 of a meter of his home, he will find food. He must eat within twelve days. The first day he crawls 1/2 meter. The second day he crawls 1/4 meter. The third day he crawls 1/8 of meter. This pattern continues for twelve days. Make a Chart that shows the distance he has covered at the end of each day and the total he has covered at the end of each day. Does he make it to the Food in time? Answered by Penny Nom. 





A project about crosses 
20030610 

From Joel:
I have this project to do about crosses and I can't think of what the answer is for the following questions: What is the area rule of the crosses (the table below will help you)? Cross Number  Area sq cm  1  5  2  13  3  25  4  41  5  61  I also need to know what the formula is for it? Answered by Penny Nom. 





Chopping trees 
20030419 

From Tamara:
The master needs some of the trees (twenty, to be exact) at the back of his spooky old mansion cleared to make way for a new evil laboratory, so he decides to send some his slaves to do the work for him. He initially sends out four of his men, armed with axes, to chop the trees down. Due to the fact he is very impatient, every ten minutes he sends out another man to help with the work. Assuming that it takes one man 30 minutes to chop down 1/3 of a tree, how long till all twenty trees are chopped down? Answered by Penny Nom. 





The sum of the first 1000 even integers 
20030206 

From Jill: What is the sum of the first 1000 even integers? Answered by Paul Betts. 





Arithmitic sequence 
20030201 

From A student: I am having problems solving this arithmetic sequence... 1, 5, 10, ___, 50, 1.00, ___, 10.00, ... I believe the answers to be 25 and 5.00 but I can't figure why. Answered by Claude Tardif. 





1+2+3+...+500 
20030131 

From Brian: What is the sum of the numbers from 1 to 500 inclusive? Answered by Paul Betts. 





A bouncing ball 
20021214 

From Eman:
Q : When a childís ball is dropped from a height h metres on to a hard, flat floor, it rebounds to a height of 3/5h metres. The ball is dropped initially from a height of 1.2m.  Find the maximum height to which the ball rises after two bounces.
 Find the total distance that the ball has traveled when it hits the floor for the tenth time.
 Assuming that the ball continues to bounce in the same way indefinitely, find the total distance that the ball travels.
Answered by Penny Nom. 





7+8+9+...+1000 
20020911 

From Shirley: My question is what is the formula for adding up numbers when you don't start with number 1? For example 3 + 4 + 5 + 6 = 18. But how could you arrive at the answer without adding all the numbers? Answered by Penny Nom. 





My salary is doubled everyday for 30 days 
20020117 

From Kanishk: I recieve 1 penny the 1st day, 2 pennies the 2nd day, and my salary is doubled everyday for 30 days. How much money will I have by the end of the 30 day time period? (Is there a way of solving this problem without a chart?) Answered by Penny Nom. 





A geometric series 
20011024 

From Tashalee: The sum of the first 3 terms of a geometric series is 13. The sum of their reciprocal is 13/9. how do you find the first three terms? Answered by Penny Nom. 





Harmonic numbers 
20010523 

From Leslie: The harmonic numbers H_{k}, k = 1,2,3.....are defined by H_{k} = 1 + 1/2 + 1/3....1/k I am trying to prove by mathematical induction: H_{2n} >= 1 + n/2 , whenever n is a nonnegative integer. H_{8} = H_{23} >= 1 + 3/2 Can you help? Answered by Harley Weston. 





A Taylor series 
20010427 

From Karan: Given the following information of the function  f''(x) = 2f(x) for every value of x
 f(0) = 1
 f(0) = 0
what is the complete Taylor series for f(x) at a = 0 Answered by Harley Weston. 





Geometric and arithmetic sequences 
20010126 

From Garry: what are the equations for geometric and arithmetic sequences? also, what are the equations for finding the sums of those series? Answered by Leeanne Boehm and Penny Nom. 





An infinite series 
20001216 

From John: summation(n=1 to infinity)[n sin(1/(2n))]^{n} Answered by Harley Weston. 





Power series representations 
20001127 

From Grace: Is there a systematic way of finding a power series representation of a function? I understand that you have to manipulate the function so that it is of the form 1/(1x), but beyond that I am lost. Answered by Harley Weston. 





Comparing an integral and a sum 
20001121 

From Douglas Norberg: A fellow teacher asked me about a problem she wanted to give to her students. It involved whether to take a million dollars or a penny doubled a number of times. I was able to determine the number must have been .01 * 2^{30} which is about $10 million and a lot more than $1 million. To check that I was right I used a spreadsheet and did a Riemann sum. When I finished I reasoned that I had done the task in several steps and I could have done it in 1 step. Thus I integrated .01 * 2^{x} from 0 through 30 but the number I got was $15,490,820.0324. Why the difference? Answered by Harley Weston. 





Infinite Geometric Series 
20001110 

From Sam Carter: I ran into a problem when studying how to find the sum of an infinite geometric series. My math book attempts to explain the concept by giving formulas involving sigma and r, but it does not really explain how to go about finding the sum of an infinite geometric series. If you could either help me with this or point me in the direction of an informative website that could help me, I'd appreciate it. Answered by Harley Weston. 





Maclaurin series again 
20000923 

From Jason Rasmussen: I suppose my confusion comes into play when I am trying to figure out where the x^{n} term comes from. I know that the Power Series notation is directly related to the Geometric Series of the form sigma [ br^{n} ] where the limit is b/(1r) for convergence at  r  <1. Therefore, the function f(x) needs to somehow take the form of b/(1(xa)), which may take some manipulation, and by setting r = (xa) and b = C_{n}, we get the Geometric Series converted to the Power Series. Taking the nth order derivative of the Power Series gives C_{n} = f^{n}(a)/n!. There must be a gap in my knowledge somewhere because I cannot seem to make f(x) = e^{x} take the form of f(x) = b/(1(xa)). Maybe I should have labelled my question as "middle" because it may be more of a personal problem with algebra and logarithms. Or, am I to assume that all functions can be represented by sigma [f^{n}(a) * (xa)^{n} / n!]? Answered by Harley Weston. 





A Maclaurin series 
20000921 

From Jason Rasmussen: I have a question regarding power series notation for certain functions within the interval of convergence. Answered by Harley Weston. 





7     77 
20000913 

From Peter: Does anyone know how to solve the following: 7 _ _ _ _ 77 ? I have to find the missing values. Answered by Chris Fisher and Walter Whiteley. 





1+4+9+16+...n^2 = n(n+1)(2n+1)/6 
20000601 

From Shamus O'Toole: How do you derive that for the series 1+4+9+16+25.. that S(n)=(n(n+1)(2n+1))/6 Answered by Penny Nom. 





Geometric sequences 
20000411 

From Jodie: I am in a grade ten principles class and was taught how to do geometric sequences and series but no one in my class understood what we were taught. Our teacher is one of few to use the new curriculum which used to be the grade twelve curriculum. Could you please explain to me how to do geometric sequences and how to find the different terms and sums. Thank you very much! Answered by Harley Weston. 





Approximations 
19991220 

From Adrian Valc: Longtime ago I red an article about surprizing (and in a way frustrating) results in math, for example the limit of a nicely defined infinite series which was believed to be a simple rational number, but later was determined to be a transcendent number that missed the rational value by an incredibly small amount (so for example the limit proved to be 2.75000...00137.. with a lot of 0's in between). I cannot find that article anymore, so i was wondering if you have any such examples, or you can point to any relevant information source? Answered by Chris Fisher. 





Radius of convergence 
19990421 

From Nowl Stave: Why is the radius of convergence of the first 6 terms of the power series expansion of x^(1/2) centered at 4 less than 6? Answered by Harley Weston. 





A Series 
19990420 

From Deepak Shrestha: Given the sequence an=e^(n*Ln(n)), does the series converge and why? Answered by Harley Weston. 





Sequences and series 
19980527 

From Michael Le Francois: The sum of the first ten terms of an arithmetic series is 100 and the first term is 1. Find the 10th term. The common ratio in a certain geometric sequence is r=0.2 and the sum of the first four terms is 1248 find the first term. Answered by Penny Nom. 

