Prove the following by induction on n

1+5+9+....+(4n-3) = n(2n-1) for all n => 1

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- Apr 19th 2008, 04:14 AMTom101Proof
Prove the following by induction on n

1+5+9+....+(4n-3) = n(2n-1) for all n => 1 - May 6th 2008, 03:27 PMTheEmptySet

check base case n=1

1=1(2(1)-1) true

assume n=k

$\displaystyle 1+5+9+....+(4k-3)=k(2k-1)$

Show k+1

$\displaystyle 1+5+9+....+(4k-3)+[4(k+1)-3]=$

by the induction hypothsis

$\displaystyle k(2k-1)+[4(k+1)-3]=2k^2-k+4k+4-3=2k^2+3k+1=$

$\displaystyle (k+1)(2k+1)=(k+1)(2(k+1)-1)$

The last one is k+1

QED