Question

If

�
(
1
)
=
9
f(1)=9 and
�
(
�
)
=
2
�
(
�
−
1
)
−
1
f(n)=2f(n−1)−1 then find the value of
�
(
5
)
f(5).

Answer 1

129

Explanation:

Given the recursion relation {f(1) = 9, f(n) = 2·f(n-1) -1}, you want the value of f(5).

Sequence

We can find the 5th term of the sequence using the recursion relation:

• f(1) = 9
• f(2) = 2·f(1) -1 = 2·9 -1 = 17
• f(3) = 2·f(2) -1 = 2·17 -1 = 33
• f(4) = 2·f(3) -1 = 2·33 -1 = 65
• f(5) = 2·f(4) -1 = 2·65 -1 = 129

The value of f(5) is 129.

__

Additional comment

After seeing the first few terms, we can speculate that a formula for term n is f(n) = 2^(n+2) +1

We can see if this satisfies the recursion relation by using it in the recursive formula for the next term.

f(n) = 2·f(n -1) -1 . . . . . . . . recursion relation

f(n) = 2·(2^((n -1) +2) +1) -1 = 2·2^(n+1) +2 -1 . . . . . using our supposed f(n)

f(n) = 2^(n+2) +1 . . . . . . . . this satisfies the recursion relation

Then for n=5, we have ...

f(5) = 2^(5+2) +1 = 2^7 +1 = 128 +1 = 129 . . . . . same as above