prove that `2+4+6+…2n=n^(2)+n`, for all natural numbers n.

1.	prove that `2+4+6+…2n=n^(2)+n`, for all natural numbers n.
Answer» Let `P(n): 2+4+6+ . . +2n=n^(2)+n` For all natural number n. Step I We observe that P(1) is true. `P(1):2=1^(2)+1` 2=2, which is true. Step II Now, assume that P(n) is true for n=k. `:.P(k):2+4+6+ . . .+2k=k^(2)+k` Step II To prove that `P(k+1):2+4+6+8 . . . +2k+2(k+1)` `=k^(2)+k+2(k+1)` `=k^(2)+2k+1+k1` `=(k+1)^(2)+k+1` So, P(k+1) is true, whenever P(k) is true. Hence, P(n) is true.

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