Question

A sequence is recursively defined by f(1)=3,f(n)=2* f(n-1) for n >=2. Write the equation for the nth term

Answer 1

Given:

$f(1)=3,f(n)=2\times f(n-1)$         ...(i)

For $n\geq 2$.

To find:

The nth term for the given recursive formula.

Solution:

The given recursive formula is of the form of

$f(n)=r\times f(n-1)$            ...(ii)

It is the recursive formula of a GP, where r is common ratio.

On comparing (i) and (ii), we get

$r=2$

Now,

First term: $a=f(1)=3$

Common difference: $r=2$

nth term of a GP is

$f(n)=ar^{n-1}$

Putting a=3 and r=2, we get

$f(n)=3(2)^{n-1}$

Therefore, the equation for the nth term is $f(n)=3(2)^{n-1}$.