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in Principle of Mathematical Induction by (46.2k points)

For all n ∈ N, by using principle of mathematical induction, then prove that:

1/2 + 1/4 + 1/8 + .... + 1/2n = 1 - 1/2n

1 Answer

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Best answer

Let the given statement is P(n), where n ∈ N,

When n = 1, then

P(1) = 1/2 = 1 - 1/21 = 1 - 1/2 = 1/2

Hence, given statement is true for n = 1, 

i.e, P(1) is true

P(m) = 1/2 + 1/4 + 1/8 + .. + 1/2m = 1 - 1/2m ......(i)

Now, we have to prove that given statement is true for n = m + 1 

i.e, P(m + 1) is also true.

Now,

Hence, the given statement is also true for n = m + 1, 

i.e., P(m + 1), is true.

Hence, by the principle of mathematical induction, we can say that the given statement is true for each natural number n ∈ N.

Hence Proved.

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