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.

Step-by-step 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=-(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=(n)/(2)(a_1+a_n)`

Then the sum of the above series is

`(26)/(2)(58-7)=13*51=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`