Question

In a geometric sequence, a4 = 54 and a7 = 1,458. what is the 12th term?

answer: B) 354,294

Answer 1

Option B:

The 12th term is 354294.

Solution:

Given data:

`a_4=54` and
`a_7=1458`

To find
`a_(12):`

The given sequence is a geometric sequence.

The general term of the geometric sequence is
`a_n=a_1\ r^(n-1)`.

If we have 2 terms of a geometric sequence
`a_n` and
`a_k` (n > K),

then we can write the general term as
`a_n=a_k\ r^(n-k)`.

Here we have
`a_4=54` and
`a_7=1458`.

So, n = 7 and k = 4 ( 7 > 4)

`a_7=a_4\ .\ r^(7-4)`

`1458=54\ . \ r^3`

This can be written as

`$r^3=(1458)/(54)`

`$r^3=27`

`$r^3=3^3`

Taking cube root on both sides of the equation, we get

r = 3

`a_(12)=a_7\ .\ r^(12-7)`

`=1458\ .\ r^5`

`=1458\ .\ 3^5`

`a_(12)=354294`

Hence the 12th term of the geometric sequence is 354294.