Question

Find the a1 and d for the arithmetic where Sn=187, n=17, and a17 = -13

Answer 1

The sum of n terms is:

`S_n=(n)/(2)(2a_1+(n-1)d)`

For n = 17, a17 = -13 and Sn = 187, then:

`\begin{gathered} 187=(17)/(2)(2a_1+(17-1)d) \\ 187\cdot(2)/(17)=2a_1+16d \\ 22\text{ = }2a_1+16d \end{gathered}`

Arithmetic sequence formula:

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

Replacing with n = 17 and a17 = -13:

`\begin{gathered} -13=a_1+(17-1)d_{} \\ -13=a_1+16d \end{gathered}`

Subtracting this equation to the previous one

`\begin{gathered} 22-(-13)=2a_1-a_1+16d-16d_{}_{}_{} \\ 35=a_1 \end{gathered}`

Then d is equal to:

-13 - 35 = 16d

-48/16 = d

-3 = d