Question
Prove the following statements by mathematical induction.
57 divides 7^(n + 2) + 8^(2n + 1), for n ∈ Z+
57 divides 7^(n + 2) + 8^(2n + 1), for n ∈ Z+
Answers
oobleck
check for n=1
7^3 + 8^3 = 855 = 57*15
assume P(k), then check for n=k+1
7^(k+1+2) + 8^(2(k+1)+1)
= 7 * 7^(k+2) + 8^2 * 8^(2k+1)
= 7 (7^(k+2) + 8^(2k+1)) + 57 * 8^(2k+1)
both terms are multiples of 57, so P(k+1)
QED
7^3 + 8^3 = 855 = 57*15
assume P(k), then check for n=k+1
7^(k+1+2) + 8^(2(k+1)+1)
= 7 * 7^(k+2) + 8^2 * 8^(2k+1)
= 7 (7^(k+2) + 8^(2k+1)) + 57 * 8^(2k+1)
both terms are multiples of 57, so P(k+1)
QED