Question

How many terms are there in a geometric series if the first term is 2, the common ratio is 4, and the sum of the series is 2,730?

Hint: cap s sub n equals start fraction a sub one left parenthesis one minus r to the power of n end power right parenthesis over one minus r end fraction comma r ≠ 1, where a1 is the first term and r is the common ratio.

n = 3
n = 4
n = 5
n = 6

Answer 1

n = 6

Explanation:

Givens

r = 4

a = 2

Sum = 2730

Equation

Sum = a(1 - r^n)/(1 - r)

Solution

2730 = 2 (1 - 4^n) / (1 - 4) Reduce the denominator

2730 = 2(1 - 4^n)/-3 Multiply both sides by -3

2730*-3 = 2(1 - 4^n) Simplify the left

-8190 = 2(1 - 4^n) Divide by 2

-8190/2 = 1 - 4^n Simplify

-4095 = 1 - 4^n Subtract 1 from both sides.

-4096 = - 4^n Divide by - 1

4096 = 4^n This looks like n is fairly large. You could use logs. Try n = 6.

4^6 does equal 4096 So your answer is 6.

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Let's try it with logs. Ignore this if you don't know how to do this.

log(4096) = log(4)^n

log(4096) = n*log(4) Divide by log 4

log(4096)/log(4) = n

3.6124 / 0.6021 = n

n = 6

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The hint is very helpful. Make sure you can follow it through