#### [recurrence relation] My solution differs from the book. Is there is error with my solution or is the book wrong?

Hanna Owens

1

Problem:
a_{n} = 2a_{n-1} + 5 and a_{0} = 3
I noticed this follows the pattern:
2^n \times a_{n} + 2^n-1 (5) + 2^n-2 (5) + ... + 2^1 (5)
which turns into
= 2^n (3) + 2^n-1 (5) + 2^n-2 (5) + ... + 2^1 (5)
and this is where my solution begins to differ from the book. If I look at:
2^n-1 + 2^n-2 + ... + 2^1
This appears to be [∑ k=1, n-1 2^k ]
So here is my question, what does this sum equal?
I believe it is 2^n - 2
So carrying on with the computations I get
2^n (3) + 5 * 2^n - 2
= 3(2^n ) + 5 * (2^n - 2)
=3(2^n ) + 5 * (2^n ) - 10
= 2^n+3 - 10
the book instead says the solution is...
= 2^n (3) + 5 * (2^n - 1)
= 3(2^n ) + 5 * (2^n ) - 5
= 2^n+3 - 5
Which is right? If I am wrong, where did I go wrong and how do I fix it?