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By using principle of mathematical induc...

By using principle of mathematical induction, prove that 2+4+6+….2n=n(n+1), `n in N`

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Let P(n): 2+4+6+…..+2n=n(n+1)
When n=1 L.H.S. =2
and R.H.S. 1(1+1)=`1xx2=2`
`therefore ` LHS=RHS, Hence P(1) is true. (a)
Let P(k) be true.
`Rightarrow 2+4+6+….+2k=k(k+1)` …..(i)
To prove P(k+1) is true i.e.
2+4+6+….2(k+1)=(k+1)(k+2) ...(ii)
Adding 2(k+1) to both sides of (i), we get
2+4+6+....2k+2(k+1)=k(k+1)+2(k+1)
=(k+1) (k+2)
Hence P(k+1) is true, whenever P(k) is true
From (a) and (b) by the principle of mathematical induction it follows that P(n) is true for all natural number n.
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