Question

1+2+2^2+_ _ _ _ +2^n=2^n+1 -1

Answer 1

Step-by-step explanation:

Let n = 1. Then:

2 + 22 + 23 + 24 + ... + 2n = 21 = 2

...and:

2n+1 – 2 = 21+1 – 2 = 22 – 2 = 4 – 2 = 2

So (*) works for n = 1.

Assume, for n = k, that (*) holds; that is, that

2 + 22 + 23 + 24 + ... + 2k = 2k+1 – 2

Let n = k + 1.

2 + 22 + 23 + 24 + ... + 2k + 2k+1

= [2 + 22 + 23 + 24 + ... + 2k] + 2k+1

= [2k+1 – 2] + 2k+1

= 2×2k+1 – 2

= 21×2k+1 – 2

= 2k+1+1 – 2

= 2(k+1)+1 – 2

Answer 2

Answer:

Step-by-step explanation:

1, 2, 2^2, 2^3, ..... , 2^n is a GP with first term 1, common ratio 2 and number of terms n+1

So Sum. = 1 x [2^(n+1)-1]/ (2-1)

= 2^(n+1)-1