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  • Let S(n) be the sum of the first n terms of the sequence 1, 3, 7, 15, ..., where each term is obtained by doubling the previous term and adding 1. Which of the followi

Let S(n) be the sum of the first n terms of the sequence 1, 3, 7, 15, ..., where each term is obtained by doubling the previous term and adding 1.

Which of the following statements is true? 

Option: 1

S(n) is an even number for all positive integers n. 


Option: 2

S(n) is an odd number for all positive integers n.


Option: 3

S(n) is an even number if n is odd, and an odd number if n is even.


Option: 4

S(n) is an odd number if n is odd, and an even number if n is even. 


Answers (1)

best_answer

We will use the principle of mathematical induction to solve this problem. 

The given sequence can be re-written as: (1, 3, 7, ..., \left( 2^{n}-\ 1 \right))

This sequence is a sequence of odd numbers.

\therefore S(n)=\left( 2^{n}-\ 1 \right) ----------(1)

When n=1, then S(1) = 1, which is an odd number.

Hence, equation (1) is true for n = 1.

Now assume that the equation is true for some natural number k.

So, S(k) is an odd number if k is odd, and an even number if k is even.

\therefore S(k)=\left( 2^{k}-\ 1 \right)----------(2)

Now, consider the sum S(k+1). 

We need to show that S(k+1) is an odd number if k+1 is odd, and an even number if k+1 is even.

This sum can be written as:

\therefore S(k+1)=\left( 2^{k+1}-\ 1 \right)

\Rightarrow (S(k+1)=\left( 2^{k}\times 2\right)-\ 1-\ 1 +1)

\Rightarrow (S(k+1)=\left( 2^{k}\times 2\right)-\ 2 +1)

\Rightarrow (S(k+1)=2\times\left( 2^{k}-\ 1\right) +1)

\Rightarrow (S(k+1)=\left( 2S(k)\right) +1)

Using the induction hypothesis, we know that S(k) is odd if k is odd, and even if k is even. Therefore, 2S(k) is even, and adding 1 to an even number gives an odd number. 

Therefore, S(k+1) is odd if S(k) is even, and even if S(k) is odd.

Hence, we can conclude that S(n) is an odd number if n is odd, and an even number if n is even.

Posted by

Gautam harsolia

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