Question

The first term in a geometric sequence is 54, and the 5th term is 2 / 3. Find an explicit form for the geometric sequence.

Answer 1

Answer:  $ar^{n-1}=54(\dfrac{1}{3})^{n-1}$

Step-by-step explanation:

The nth term for a geometric sequence is given by :-

$a_n=ar^{n-1}$    (1)

We are given that The first term in a geometric sequence is 54. i.e. a=54

5th term=$\dfrac{2}{3}$   (2)

Put n=5 and a= 54 in (1), we get

$a_5=(54)r^{4}=$     (3)

From (2) and (3), we have

$\Rightarrow(54)r^{4}=\dfrac{2}{3}\\\\\Rightarrow r^4=\dfrac{2}{3}\times\dfrac{1}{54}=\dfrac{1}{3\times27}=\dfrac{1}{81}\\\\\Rightarrow\ r=(\dfrac{1}{81})^{\frac{1}{4}}=\dfrac{1}{3}$

Explicit form for the geometric sequence: $ar^{n-1}=54(\dfrac{1}{3})^{n-1}$