Chapter 8.2, Problem 68E

To determine

To find:The12th partial sum of given arithmetic sequence.

Answer to Problem 68E

The12th partial sum of the given arithmetic sequence is $17.4$ .

Explanation of Solution

Given information:

An arithmetic sequence is given as $4.2,3.7,3.2,2.7,\mathrm{...}$

Concept used:

An arithmetic sequence of n terms, has the form $a_1,a_2,a_3,\mathrm{...},a_n$ and should have a common difference between each two consecutive terms.

That is

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}$

Common difference can be defined by d .

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}=d$

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.

Sum of an arithmetic finite sequence has the form

  $S_n=\frac{n}{2}\left(a_1+a_n\right)$

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

Calculation:

Consider the given sequence.

  $4.2,3.7,3.2,2.7,\mathrm{...}$

Now, first term of the finite sequence is $a_1=4.2$ .

Common difference is

  $\begin{array}{c}d=a_2-a_1\\ =3.7-4.2\\ =-0.5\end{array}$

12th term of the sequence can be found as

. $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ a_{12}=4.2+\left(12-1\right)\left(-0.5\right)\\ =4.2-5.5\\ =-1.3\end{array}$

Number of term for the required sumis indicated as 12.

So, the 12th partial sum is calculated as

  $\begin{array}{c}{S}_n=\frac{n}{2}\left(a_1+a_n\right)\\ S_{12}=\frac{12}{2}\left(4.2-1.3\right)\\ =6\times 2.9\\ =17.4\end{array}$

Thus, the 12th partial sum of the given arithmetic sequence is $17.4$ .