Chapter 8.2, Problem 61E

To calculate: The sum of the arithmetic sequence of the nth partial for $0,-9,-18,-27,\mathrm{...},n=40$

integer from −10 to 50.

To determine

The sum of the partial arithmetic sequence $0,-9,-18,-27,\mathrm{...},n=40$ is $-7020$ .

Given information:

The given sequence is $0,-9,-18,-27,\mathrm{...},n=40$ .

Definition used:

The nth term of the arithmetic sequence has the form $a_n=a_1+\left(n-1\right)d$ ,where $a_1$ is the first term of the sequence, and d is the common difference.

The sum of finite arithmetic sequence is given by $S_n=\frac{n\left(a_1+a_n\right)}{2}$ .

Here n is the number of terms, $a_1$ is the first term of the sequence, and $a_n$ is the last tems of sequence.

Calculation:

Compute the sum of partial arithmetic sequence.

The nth term of the arithmetic sequence has the form $a_n=a_1+\left(n-1\right)d$ .

Substitute 40 for n and 0 for $a_1$ and −9 for d in the above formula as follows,

  $\begin{array}{c}{a}_{40}=0+\left(40-1\right)\left(-39\right)\\ =0+39\left(-9\right)\\ -351\end{array}$

Compute the sum of finite arithmetic sequence as follows,

  $\begin{array}{c}{S}_n=\frac{40\left(0+\left(-351\right)\right)}{2}\\ =-7020\end{array}$

Therefore, the sum of the partial arithmetic sequenceis $-7020$ .

Answer to Problem 61E

The sum of the partial arithmetic sequence $0,-9,-18,-27,\mathrm{...},n=40$ is $-7020$ .

Explanation of Solution

Given information:

The given sequence is $0,-9,-18,-27,\mathrm{...},n=40$ .

Definition used:

The nth term of the arithmetic sequence has the form $a_n=a_1+\left(n-1\right)d$ ,where $a_1$ is the first term of the sequence, and d is the common difference.

The sum of finite arithmetic sequence is given by $S_n=\frac{n\left(a_1+a_n\right)}{2}$ .

Here n is the number of terms, $a_1$ is the first term of the sequence, and $a_n$ is the last tems of sequence.

Calculation:

Compute the sum of partial arithmetic sequence.

The nth term of the arithmetic sequence has the form $a_n=a_1+\left(n-1\right)d$ .

Substitute 40 for n and 0 for $a_1$ and −9 for d in the above formula as follows,

  $\begin{array}{c}{a}_{40}=0+\left(40-1\right)\left(-39\right)\\ =0+39\left(-9\right)\\ -351\end{array}$

Compute the sum of finite arithmetic sequence as follows,

  $\begin{array}{c}{S}_n=\frac{40\left(0+\left(-351\right)\right)}{2}\\ =-7020\end{array}$

Therefore, the sum of the partial arithmetic sequenceis $-7020$ .