Chapter 11.6, Problem 2ST

a.

To determine

To calculate:

Sum of given arithmetic series.

Answer to Problem 2ST

Sum of given arithmetic series is $1080$ .

Explanation of Solution

Give information:

  $n=30,t_1=7,t_{30}=65$

Calculation:

In order to find out sum of arithmetic series,

Consider,

  $n=30,t_1=7,t_{30}=65$

Sum of arithmetic series is calculated by,

  $\frac{n}{2}\left(a+l\right)$

Where,

  $a=$ initial term,

  $l=$ last term,

  $n=$ number of terms.

Sum of arithmetic series having $n$ terms in which first term $a$ and last term is $l$ is,

  $\frac{n}{2}\left(a+l\right)$

Put the values in above equation,

  $\begin{array}{l}\frac{n}{2}\left(a+l\right)\\ =\frac{30}{2}\left(7+65\right)\\ =15\left(72\right)\\ =1080\end{array}$

Therefore, Sum of given arithmetic series is $1080$ .

b.

To determine

To calculate:

Sum of given arithmetic series.

Answer to Problem 2ST

Sum of given arithmetic series is $-45$ .

Explanation of Solution

Give information:

  ${\displaystyle \sum _{k=1}^{15}\left(45-6k\right)}$

Calculation:

In order to find out sum of arithmetic series,

Consider,

  ${\displaystyle \sum _{k=1}^{15}\left(45-6k\right)}$

Sum of arithmetic series is calculated by,

  $\frac{n}{2}\left(a+l\right)$

Where,

  $a=$ initial term,

  $l=$ last term,

  $n=$ number of terms.

Sum of arithmetic series having $n$ terms in which first term $a$ and last term is $l$ is,

  $\frac{n}{2}\left(a+l\right)$

Here $a=39,n=15,l=-45$

Put the values in above equation,

  $\begin{array}{l}\frac{n}{2}\left(a+l\right)\\ =\frac{15}{2}\left(39+\left(-45\right)\right)\\ =7.5\left(-6\right)\\ =-45\end{array}$

Therefore, Sum of given arithmetic series is $-45$ .