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

$r^3=\frac{1458}{54}$

$r^3=27$

$r^3=3^3$

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.