1 2 22 2n 2n 1 1 For All Natural Numbers

1 + 2 + 22 + … + 2n = 2n +1 – 1 for all natural numbers .

Answers:

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.