Answer to Question #186556 in Discrete Mathematics for Maricel

Question

Solve the following. (10 pts each)

1. Prove P(n) = n2(n + 1)

2. Recurrence relation an = 2n with the initial term a1= 2.

Answer 1

(1) $P(n)=n^2(n+1)\\$

$= n^3+n \\$

$S_n= \sum P(n)=\sum n^3+\sum n$

$= (\dfrac{n(n+1)}{2})^2 + \dfrac{n(n+1)}{2}\\ \\ =\dfrac{n^4+n^3+2n^2+n^3+n^2+2n}{4}\\=\dfrac{1}{4}n(n+1)(n^2+n+2)$

(2) $a_n=2n \ \ \ \ \ \ a_1=2$

This is already a recurrence relation with sum

$S_n=\sum 2n=2\dfrac{n(n+1)}{2}=n(n+1)$