Subject: geometric sequences

Name: Jodie
Who is asking: Student
Level: Secondary

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!

Hi Jodie,

The distinguishing feature of geometric sequences is that there is a number r, called the common ratio, so that each term in the sequence (after the first term) is r times the previous term. For example

5, 15, 45, 135, 405

is a geometric sequence since each term, after the first, is 3 times the previous term. Also

1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64

is a geometric series since each term, after the first is 1/2 times the previous term. However

1, 1/2, 1/3, 1/4, 1/5

is not a geometric series since the second term is 1/2 times the first term but the third term is 2/3 times the second term.

To write this algebraically, if the first term is a and the common ratio is r then the first n terms of the geometric sequence are

a, a r, a r2, a r3, a r4, ..., a rn-1

Notice that the n-th term is a rn-1 not a rn since the first term is a = a r0.

Hence if you have a geometric sequence with a = 7 and r = 1/2 then the 6th term is

a r5 = 7 (1/2)5 = 7/32.

To see how to find the sum of a geometric series I want to extend my first example and find

S = 5 + 15 + 45 + 135 + 405 +...+ 98415 + 295245

The technique is to multiply both sides of the equation by the common ratio 3 to get the two equations

S =  5 + 15 + 45 +...+ 98415 + 295245
3S =        15 + 45 +...+ 98415 + 295245 + 885735

(I have shifted the second equation to the right to line up equal terms.)

Now subtract the first equation from the second to get

3S - S = 885735 - 5

And thus

S = 885730/2 = 442865

Your teacher may have showed you a formula to find the sum of the first n terms of a geometric series but I think it is more important to understand the technique of multiplying both sides by the common ratio and subtracting. This is the technique that leads to the formula.

I hope this helps,
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