Chapter 12.1, Problem 55E

a.

To determine

To calculate: The sum of first 5 terms of a sequence of odd natural numbers.

Answer to Problem 55E

The sum of first 5 terms of a sequence of odd natural numbersis $25$ .

Explanation of Solution

Given information:

The sequence of odd natural numbers.

Formula used:

A sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

The sum of n terms of the arithmetic sequence is $S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$ .

Calculation:

Consider the sequence of odd natural numbers.

The arithmetic series is $1+3+5+7+9$ .

Recall that a sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

Compute the difference between the consecutive terms,

  $\begin{array}{c}{a}_2-a_1=3-1\\ =2\end{array}$

And

  $\begin{array}{c}{a}_3-a_2=5-3\\ =2\end{array}$

Therefore, common difference is $2$ .

According to question $a_1=1$ and $n=5$ .

The sum of n terms of the arithmetic sequence is $S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$ .

Apply it,

  $\begin{array}{c}{S}_5=\frac{5}{2}\left[2\left(1\right)+\left(5-1\right)\left(2\right)\right]\\ =\frac{5}{2}\left[2+8\right]\\ =\frac{5}{2}\left(10\right)\\ =25\end{array}$

Thus, the sum of first 5 terms of a sequence of odd natural numbersis $25$ .

b.

To determine

To calculate: The sum of first 10 terms of a sequence of odd natural numbers.

Answer to Problem 55E

The sum of first 10 terms of a sequence of odd natural numbersis $100$ .

Explanation of Solution

Given information:

The sequence of odd natural numbers.

Formula used:

A sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

The sum of n terms of the arithmetic sequence is $S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$ .

Calculation:

Consider the sequence of odd natural numbers.

The arithmetic series is $1+3+5+7+9+\dots$ .

Recall that a sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

Compute the difference between the consecutive terms,

  $\begin{array}{c}{a}_2-a_1=3-1\\ =2\end{array}$

And

  $\begin{array}{c}{a}_3-a_2=5-3\\ =2\end{array}$

Therefore, common difference is $2$ .

According to question $a_1=1$ and $n=10$ .

The sum of n terms of the arithmetic sequence is $S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$ .

Apply it,

  $\begin{array}{c}{S}_{10}=\frac{10}{2}\left[2\left(1\right)+\left(10-1\right)\left(2\right)\right]\\ =5\left[2+18\right]\\ =5\left(20\right)\\ =100\end{array}$

Thus, the sum of first 10 terms of a sequence of odd natural numbersis $100$ .

c.

To determine

To calculate: The sum of first n terms of a sequence of odd natural numbers.

Answer to Problem 55E

The sum of first n terms of a sequence of odd natural numbers is $n^2$ .

Explanation of Solution

Given information:

The sequence of odd natural numbers.

Formula used:

A sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

The sum of n terms of the arithmetic sequence is $S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$ .

Calculation:

Consider the sequence of odd natural numbers.

The general form of an odd natural number is $\left(2n-1\right)$ .

The arithmetic series is $1+3+5+7+9+\dots +\left(2n-1\right)$ .

Recall that a sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

Compute the difference between the consecutive terms,

  $\begin{array}{c}{a}_2-a_1=3-1\\ =2\end{array}$

And

  $\begin{array}{c}{a}_3-a_2=5-3\\ =2\end{array}$

Therefore, common difference is $2$ .

According to question $a_1=1$ and $n=n$ .

The sum of n terms of the arithmetic sequence is $S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$ .

Apply it,

  $\begin{array}{c}{S}_n=\frac{n}{2}\left[2\left(1\right)+\left(n-1\right)\left(2\right)\right]\\ =\frac{n}{2}\left[2+2n-2\right]\\ =\frac{n}{2}\left(2n\right)\\ =n^2\end{array}$

Thus, the sum of first n terms of a sequence of odd natural numbers is $n^2$ .