Chapter 9.5, Problem 39E

Solution

To identify : The correct option for the statement “In a proof by mathematical induction that $1+2+3+\dots +n=\frac{n\left(n+1\right)}{2}$ , the inductive hypothesis would be to assume among the following options”.

Option (A) $n=1$

Option (B) $k=1$

Option (C) $1=\frac{1\left(1+1\right)}{2}$

Option (D) $1+2+3+\dots +n=\frac{n\left(n+1\right)}{2}$

Option (E) $1+2+3+\dots +k=\frac{k\left(k+1\right)}{2}$

Option (E) is correct option.

Given information :

The statement $1+2+3+\dots +n=\frac{n\left(n+1\right)}{2}$ .

Explanation :

Consider the statement $1+2+3+\dots +n=\frac{n\left(n+1\right)}{2}$ .

To prove that it is true by mathematical induction, the following steps are followed.

Anchor step: Prove that any statement $P_n$ is true for $n=1$ .

Inductive hypothesis: Assume $P_n$ is true for $n=k$ .

Inductive Step: Prove that $P_n$ is true for $n=k+1$ .

So, at the inductive hypothesis it is assumed that $P_n$ holds true for $n=k$ .

  $1+2+3+\dots +k=\frac{k\left(k+1\right)}{2}$

So, the option (E) is correct.