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Question

Use the principle of mathematical induction to show that :
2+22+.....+2n=2n+12 for every natural number n.

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Solution

p(n):2+22+..............+2n=2n+12When,n=1L.H.S=2andR.H.S=2so,forn=1p(n)istrue.LetP(R):2+22+....2k=2k+12Therefore,p(k+1):2+22+....2k+1=2k+22=2k+1.22=2(2k+11)So,p(k+1)isalsotrue.Hencebytheprincipleofmathematicalinduction,p(n)istrueforallnN.

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