If a and b are integers and the sequence a_0, a_1, a_2, ... and b_0, b_1, b_2, ... are defined as a_{n+1} = a_n + b_n an...

Question

If a and b are integers and the sequence a_0, a_1, a_2, ... and b_0, b_1, b_2, ... are defined as a_{n+1} = a_n + b_n and b_{n+1} = a_n - b_n for n >= 0, where a_0, b_0, m are natural numbers, what is the value of (a_m + b_m + a_{m+1} + b_{m+1})?

A. 2a(k+1)   B. a(k+1)   C. a(k+1) - 1   D. a^2 + b^2

Accepted Answer

B. The sum a_m + b_m + a_{m+1} + b_{m+1} simplifies to a(k+1) in the notation of the problem. We have two interleaved integer sequences where each term of one is built by adding or subtracting the previous terms of both. The question asks us to simplify the sum of four consecutive values, which turns out to reduce to a neat expression involving initial values. Recognising the recurrence as a linear transformation lets us track how the sum changes at each step.