Question

Evaluate arithmetic series:- Step-by-step answer, please!

Answer 1

Answer:

-186,450

Step-by-step explanation:

Sum of arithmetic series formula

$S_n=\dfrac{n}{2}[2a+(n-1)d]$

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:

• $a=7$
• $d=-5$
• $n=275$

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

Answer 2

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:-

$\\ \rm\Rrightarrow S_n=\dfrac{n}{2}[a+l]$

$\\ \rm\Rrightarrow \dfrac{275}{2}(7-1363)$

$\\ \rm\Rrightarrow \dfrac{275}{2}(-1356)$

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

$\\ \rm\Rrightarrow -186450$