Proof By Induction Series Questions, Answers and Solutions

  a - answer    s - solution    v - video

Question 1

Prove by mathematical induction for $n\ge1$ and $n\in\mathbb{Z}^+$:

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

b) $1+3+5+7+\dotsb+(2n-1)=n^2$a s v

c) $1+4+7+10+\dotsb+(3n-2)=\frac{n(3n-1)}{2}$a s v

d) $1^2+2^2+3^2+4^2+\dotsb+n^2=\frac{n(n+1)(2n+1)}{6}$a s v

e) $2^1+2^2+2^3+2^4+\dotsb+2^n=2^{n+1}-2$a s v

f) $\frac{1}{1\times2}+\frac{1}{2\times3}+\dotsb+\frac{1}{n(n+1)}=\frac{n}{n+1}$a s v

g) $\frac{1}{4\times1^2-1}+\frac{1}{4\times2^2-1}+\dotsb+\frac{1}{4\times n^2-1}=\frac{n}{2n+1}$a s v

h) $1^3+2^3+3^3+\dots+n^3=(1+2+3+\dots+n)^2$a s v

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