Question

Find the Arithmetic series and the sum of the series for which a1=58 an=-7 and n=26

Answer 1

Given:

$a_1=58,\text{ }a_n=-7,\text{ and }n=26$

Required:

We have to find the arithmetic series.

Explanation:

From the given data if we can find the common difference denoted by d then we can easily find the required arithmetic series.

We use the formula

$a_n=a_1+(n-1)d$

Now we put the given values in the above equation to find the value of d.

$\begin{gathered} -7=58+(26-1)d \\ \Rightarrow-7-58=25d \end{gathered}$$\begin{gathered} \Rightarrow25d=-65 \\ \\ \Rightarrow d=-\frac{65}{25} \end{gathered}$$\Rightarrow d=-2.6$

Then the required arithmetic series is

$a_1,a_1+d,a_1+2d,a_1+3d,.\text{ }.\text{ }.\text{ },a_1+25d$$\begin{gathered} =58,\text{ }58-2.6,\text{ }58-5.2,\text{ }58-7.8,\text{ }.\text{ }.\text{ }.,\text{ }58-65 \\ =58,\text{ }55.4,\text{ }52.8,\text{ }50.2,\text{ }.\text{ }.\text{ }.\text{ },-7 \end{gathered}$

The formula for finding the sum of the arithmetic series is

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

Then the sum of the above series is

$\frac{26}{2}(58-7)=13\times51=663$

Final answer:

Hence the arithmetic series is

$\begin{equation*} 58,\text{ }55.4,\text{ }52.8,\text{ }50.2,\text{ }.\text{ }.\text{ }.\text{ },-7 \end{equation*}$

And the sum of the series is

$663$