Question

Evaluate arithmetic series:-

Step-by-step answer, please!

Answer 1

Let's see

`\\ \rm\Rrightarrow {\displaystyle{\sum^(275)_(k=1)}}(-5k+12)`

`\\ \rm\Rrightarrow (-5(1)+12)+(-5(2)+12)\dots (-5(275)+12)`

`\\ \rm\Rrightarrow 7+5+3+2+1+\dots -1363`

So

• a=7
• l=-1363
• n=275

Sum:-

`\\ \rm\Rrightarrow S_n=(n)/(2)[a+l]`

`\\ \rm\Rrightarrow (275)/(2)(7-1363)`

`\\ \rm\Rrightarrow (275)/(2)(-1356)`

`\\ \rm\Rrightarrow 275(-678)`

`\\ \rm\Rrightarrow -186450`

Answer 2

-186,450

Explanation:

Sum of arithmetic series formula

`S_n=(n)/(2)[2a+(n-1)d]`

where:

• a is the first term
• d is the common difference between the terms
• n is the total number of terms in the sequence

`\displaystyle \sum\limits_(k=1)^(275) (-5k+12)`

To find the first term, substitute
`k = 1` into
`(-5k+12)`

`a_1=-5(1)+12=7`

To find the common difference, find
`a_2` then subtract
`a_1` from
`a_2`:

`a_2=-5(2)+12=2`

`\begin{aligned}d & =a_2-a_1\\ & =2-7\\ & =-5\end{aligned}`

Given:

• 
  `a=7`
• 
  `d=-5`
• 
  `n=275`

`\begin{aligned}S_(275) & = (275)/(2)[2(7)+(275-1)(-5)]\\& = (275)/(2)[14-1370]\\& = (275)/(2)[-1356]\\& = -186450\end{aligned}`