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

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

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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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