Geometric Series

Question

1. How many terms of the geometric series 3+3+3+...would add up to 528?

2. In a geometric series, of Ss = 44 and r=-2, find a1

3. Find the first term of a geometric sequence whose last term is 1458, common ratio is -3 and the sum all the terms is 1094.

4. In a geometric series, Sn = -1364, r = -2 and a1 =4, find n.

Answer 1

Hi, Ryan!

Your answers, in order:

1. If I'm reading this correctly, this is just a sum. If the starting number of the series is 3, and the common ratio is 1, then you can solve this problem by simple division. This, however, is odd to me, as normally a geometric sequence should not have a common ratio of 1. Am I reading this correctly? If so, 528/3 = 176.
2. Was "Ss" a typo for S₅ on your teacher's part? If so, here's the answer. If not, feel free to let me know and I'll try it again.

For a geometric series where S₅ is 44, we can use the following equation:

S_n= (a₁*(1-rⁿ))/(1-r)

At this point, we need to solve for a1.

(S_n*(1-r))/(1-rⁿ) = a₁

Plugging in our values of Sn = 44, r = -2, and n = 5, we get that

a₁ of this series equals 4.

1. If we reverse-engineer this series, we could easily say that the sequence has an a₁ of 1458 and a common ratio of -1/3, as each number will get smaller if we go in reverse order. The equation for finding out the sum of the first n numbers of a geometric series is

S_n= (a₁*(1-rⁿ))/(1-r)

At this point, we need to solve for n.

-((S_n*(1-r))/a₁)+1 = rⁿ

This would mean that r^n = -0.0004572473708275293

r is equal to -1/3. For ease of calculation, let's multiply that number by negative one, and then take the log_1/3 of this number.

log_1/3(0.0004572473708275293) = 7.

So there are 7 numbers in this series. You could use this same formula to derive this, but we could also just divide it one-by-one to demonstrate what we have.

Our series: 1458,-486,162,-54,18,-6,2.

To test this, we can add these back up one-by one.

2-6+18-54+162-486+1458 = 1094.

Sure enough, the answer to this one is 2.

1. Here we get to use our previously-solved equation, which will make things a bit quicker.

S_n = -1364

r = -2

a1 = 4

-((S_n*(1-r))/a₁)+1 = rⁿ

We have our formula. Solving for r^n gives us 1024.

log₂1024 = 10.

Brute-force addition will confirm this. The series in question is [4, -8, 16, -32, 64, -128, 256, -512, 1024, -2048]. Summing these together gives us -1364

n = 10