Mathematical Induction

**oldguy** · November 21st 2007, 06:29 PM

Question:

Use the principal of Mathematical Induction to prove 2|(n2+n) for all n>=0

**kalagota** · November 22nd 2007, 06:13 AM

Originally Posted by oldguy

Question:

Use the principal of Mathematical Induction to prove 2|(n2+n) for all n>=0

**Soroban** · November 22nd 2007, 06:21 AM

Hello, oldguy!

Use Mathematical Induction to prove: . $2\,|\,(n^2+n)$ for all $n \geq 0$

$S(n)\!: \;n^2+n$ is a multiple of 2.

Verify $S(1)\!:\;\;1^2 + 1 \:=\:2$ . . . True!

Assume $S(k)\!:\;\;k^2 + k \:=\:2a$ for some integer $a$

Add $2k+2$ to both sides: . $k^2 + k + {\color{blue}2k + 2} \;=\;2a + {\color{blue}2k + 2}$

We have: . $k^2 + 2k + 1 + k + 1 \;=\;2a + 2k + 2$

. . . . . . . .. $\underbrace{(k+1)^2 + (k+1)}_{\text{Left side of S(k+1)}} \;=\;\underbrace{2(a + k + 1)}_{\text{multiple of 2}}$

Therefore, we have proved $S(k+1).$
. . The inductive proof is complete.

**oldguy** · November 22nd 2007, 06:56 AM

Thanks for the help.

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