Answer to Question #117408 in Algebra for Nikhil

Question

Find the sum of finite geometric series
: 2+1/2 + 1/4 +1/8 ............+1/2^(n-1)

Answer 1

1. We say that S(n) = 3- 1/2^(n-1). Show that this formula by help of mathematical induction is true in case n=1 and if n = k+1 ;
2. S(1) = 2 ; S(1) = 3- 1/2^(1-1) = 3 - 1 = 2
3. n = k + 1. S(k+1) = 2+1/2+1/4 +..+1/2^k + 1/2^k-1= S(k) + 1/2^k
4. S(k) = 3- 1/2^(k-1). Then S(k+1) = S(k) + 1/2^k = 3 - 1/2^(k-1)+1/2^k =

= 3 - 1/(2^k * 2^-1) + 1/2^k = 3 - 2/2^k + 1/2^k = 3 - 1/2^k;

   5.If we show to the first formula and substitute n = k+1, S(k+1) = 3- 1/2^(k+1-1) =
= 3 - 1/2^k
   6. Saw that if the formula is true for n=k, then it is true for n = k+ 1,
thus, the formula is true for any n;