Geometric series

Question

The first and the last terms of a geometric series are 2 and 2048 respectively. The sum of the series is 2730. Find the number of terms and the common ratio.

Answer 1

Let the terms be 2,2x,2x²,…,2x^n-1.

 

Then we have the following system:

 

1. 2x^n-1 = 2048

 

2. 2(xⁿ - 1)/(x - 1) = 2730

 

The first equation becomes xⁿ = 1024x.

 

Substitute that in the second equation and solve for x:

 

2(1024x - 1)/(x - 1) = 2730

 

1024x - 1 = 1365(x - 1)

 

1024x - 1 = 1365x - 1365

 

1364 = 341x

 

4 = x

 

Finally, plug this in and find n.

 

4ⁿ = 4096

 

n = 6

 

So the sequence is 2,8,32,128,512,2048 with 6 terms and common ratio 4.