Chapter 11.7, Problem 38HP

To determine

Write a sequence, the formula that produces it, and determine the formula for the sum of the terms of the sequence. Then prove the formula with mathematical induction.

Answer to Problem 38HP

Sequence: $1,2,3,4,\mathrm{......}n$ ; formula $a_n=n$

Explanation of Solution

Given:

Write a sequence, the formula that produces it, and determine the formula for the sum of the terms of the sequence. Then prove the formula with mathematical induction.

Concept Used:

Sequence: $1,2,3,4,\mathrm{......}n$

Calculation:

Sequence: $1,2,3,4,\mathrm{......}n$

Formula:

  $\begin{array}{l}{a}_1=1;d=2-1=1\\ a_n=a_1+(n-1)·d=1+(n-1)·1=1+n-1=n\\ a_n=n\end{array}$

Sum of $1,2,3,4,\mathrm{......}n=\frac{n(n+1)}{2}$

We first check the base case, n = 1. The sum is 1 and the formula evaluates to 1 as well, so we are good.

Induction hypothesis: Assume that for some $n\ge 1$ we have: ${\displaystyle \sum _{k=1}^{n}k}=\frac{n(n+1)}{2}$

We add (n+1) on both sides of this relation and get ${\displaystyle \sum _{k=1}^{n+1}k}=\frac{n(n+1)}{2}+(n+1)$

The right hand side can now rewritten as ${\displaystyle \sum _{k=1}^{n+1}k}=\frac{[n+1]([n+1]+1)}{2}$

Thus, we have proved that if the formula holds for n, it holds for n+1. By the principle of mathematical induction, ${\displaystyle \sum _{k=1}^{n}k}=\frac{n(n+1)}{2}$ the sum of all natural number is true.

Thus, the Sequence: $1,2,3,4,\mathrm{......}n$ ; formula $a_n=n$