Question

2+4+6+.....+2n=n ka square +n solve by pmi statement​

Answer 1

Answer:

let p(n) =2+4+6+........+2n =n^2+n

p(n)=2(1+2+3+..............+n) =n^2+n

put n=1

p(1)=2(1+2+3+.......+1)=1^2 +1

2=2

p(1)=2

p(1)is true

Assume that the statement p(k) is true

(ie) p(k)=2(1+2+3+..........+k) =k^2+k

We need to show that p(k+1)is true

p(k+1)=2(1+2+3+.............+k+k+1)

=k^2+k+2(k+1)

=k(k+1)+2(k+1)

=(k+1)(k+2)

=(k+1)(k+1+1)

=(k+1)^2 + (k+1)

p(k+1)=(k+1)^2 +(k+1)

p(k+1) is true

therefore by the principle of mathematical induction p(n):=2+4+6+..........+2n=n^2+n