Question

Given that the sum of the first n terms of the provided series is 6560 determine the value of n (2,6,18,54....)

Answer 1

n = 8

Explanation:

The given sequence, 2, 6, 18, 54. . ., is a geometric sequence.

It has a common ratio of 3 =>
`(6)/(2) = (18)/(6) = (54)/(18) = 3`

Thus, the sum of the first n terms of a geometric sequence is given as
`S_n = (a_1(1 - r^n))/(1 - r)`

Where,

`a_1` = first term of the series = 2

r = common ratio = 3

`S_n` = sum of the first n terms = 6,560

Plug in the above values into the formula

`6,560 = (2(1 - 3^n))/(1 - 3)`

`6,560 = (2(1 - 3^n))/(-2)`

`6,560 = (1 - 3^n)/(-1)`

Multiply both sides by -1

`-6,560 = 1 - 3^n`

Subtract 1 from both sides

`-6,560 - 1 = - 3^n`

`-6,561 = - 3^n`

`6,561 = 3^n`

Evaluate

`3^8 = 3^n`

3 cancels 3

`8 = n`

The value of n = 8