Question

1 + 2 + 2^2 + … + 2^n = 2^(n +1) – 1 for all natural numbers n.​

Answer 1

Step-by-step explanation:

Let P(n): 1 + 2 + 22 + … + 2n = 2n +1 – 1, for all natural numbers n

P(1): 1 =20 + 1 - 1 = 2 - 1 = 1, which is true.

Hence, ,P(1) is true.

Let us assume that P(n) is true for some natural number n = k.

P(k): l+2 + 22+…+2k = 2k+1-l (i)

Now, we have to prove that P(k + 1) is true.

P(k+1): 1+2 + 22+ …+2k + 2k+1

= 2k +1 – 1 + 2k+1 [Using (i)]

= 2.2k+l– 1 = 1

= 2(k+1)+1-1

Hence, P(k + 1) is true whenever P(k) is true.

So, by the principle of mathematical induction P(n) is true for any natural number n. .

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