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Prove that 1+2+2^(2)+ . . .+2^(n)=2^(n+1...

Prove that `1+2+2^(2)+ . . .+2^(n)=2^(n+1)-1`, for all natural number n.

Text Solution

Verified by Experts

Consider the given statement
`P(n):1+2+2^(2)+ . . .2^(n)=2^(n+1)-1`, for natural numbers n.
Step I We observe that P(0) is true.
`P(1):1=2^(0+1)-1`
`1=2^(1)-1`
1=2-1
1=1, which is true.
Step II Now, assume that P(n) is true for n=k.
So, P(k) : `1+2+2^(2)+ . . .2^(k)=2^(k+1)-1` is true.
Step III Now, to prove P(k+1) is true.
`P(k+1):1+2+2^(2)+. . .+2^(k)+2^(k+1)`
`=2^(k+1)-1+2^(k+1)`
`=2*2^(k+1)-1`
`=2^(k+1)+1-1`
So, P(k+1) is true, whenever P(k) is true.
Hence, P(n) is true.
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