(a) Give a recursive definition of the set P of all non negative integers,(adsbygoogle = window.adsbygoogle || []).push({});

(b) formulate the applicable induction principle and

(c) then apply the induction principle to prove that 1/2^0+1/2^1+1/2^2....+1/2^i = 2-1/2^n for n>=0

I have solved parts a and b and stuck on c

(a) P is the smallest subset of R (Real numbers) such that 0 belongs to P and if k belongs to P then also k+1 belongs to P. Recursive definition

(b) If a subset B of P is such that 0 belongs to B and if k belongs to B then also k+1 belongs to B, then subset B is equal to P. Induction principle

(c) Proof:

Step 1:

Let B = {nā n belongs to P, 1/2^0+1/2^1+1/2^2ā¦+1/2^n = 2-1/2^n}

Step 2:

0 belongs to B: 0 belongs to B because 1/2^0 =2- 1/2^0 Therefore 1 = 1

Step 3:

Let k belong to B, thus 1/2^0+1/2^1+1/2^2ā¦+1/2^k = 2-1/2^k

Is k+1 belong to B? I am stuck Here

Any ideas?

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# Mathematical induction

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