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

Answers (1)

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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