Proof By Induction Sigma Notation Questions, Answers and Solutions

  a - answer    s - solution    v - video

Question 1

Prove by mathematical induction for $n\in\mathbb{Z}^+$:

a) $\sum_{x=1}^n x^3 = \frac{n^2{(n+1)}^2}{4}$a s v

b) $\sum_{x=1}^n x(x+1) = \frac{n(n+1)(n+2)}{3}$a s v

c) $\sum_{x=1}^n \frac{1}{x(x+1)} = \frac{n}{n+1}$a s v

d) $\sum_{x=1}^n x\times x! = (n+1)!-1$a s v

e) $\sum_{x=0}^n r^x = \frac{1-r^{n+1}}{1-r}$, $r\neq1$a s v

f) $\sum_{x=0}^n ar^x = \frac{a(1-r^{n+1})}{1-r}$, $r\neq1$a s v

g) $\sum_{x=1}^n (-1)^x\times x^2 = \frac{(-1)^nn(n+1)}{2}$a s v

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