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Prove by induction that 1+5+9+......+(4n-3)=n(2n-1),for all n?N

Answers (1)

D Deependra Verma

Let P(n) be the statement
$1+5+9+......+(4n-3)=n(2n-1)$
Putting the value as P(1),
$1=1(2.1-1)i.e.,, 1=1,which, is, trueRightarrow P(1), is, true$
Let P(m) be also true,
i.e., $1+5+9+......+(4m-3)=m(2m-1)$ ............(i)
For $P(m+1)$ to be true
$P(m+1):1+5+9+......+(4m-3)+(4(m+1)-3)$
$m(2m-1)+(4m+1)$ (using(i))
$2m^2+3m+1=(m+1)(2m+1)$
$(m+1)(2(m+1)-1)$
$Rightarrow P(m+1), is, true$
Hence,by induction,P(n) is true for all n∈N.

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