prove that 1+5+9+ . . .+(4n-3)=n(2n-1), for all natural number n.

Explanation

Let P(n):1+5+9+ . . .+(4n-3)=n(2n-1), for all natural number n.
Step I We observe that P(1) is true.
`P(1):1=1(2xx1-1), 1=2-1` and 1=1, which is true.
Step II Now assume that P(n) is true for n=k.
So, P(k):1+5+9+ . . .+(4k-3) = k(2k-1) is true.
Step III Now, to prove P(k+1) is true.
(P(k+1):1+5+9+. . . +(4k-3)+4k+1)-3
=k(2k-1)+4(k+1)-3
`=2k^(2)-k+4k+4-3`
`=2k^(2)+3k+1`
`=2k^(2)+2k+k+1`
2K(k+1)+1(k+1)
=(k+1)(2k+1)
=(k+1)[2k+1+1-1]
=(k+1)[2(k+1)-1]
So, P(k+1) is true, whenever p(k) is true, hence p(n) is true.