Question

question 4 · 1 point
evaluate the following sum.
\\(\sum_{k = 4}^{13}(5k + 8)\\)
select the correct answer below:
555
505
454
204
50

Explanation:

Step1: Split the sum

We know that $\sum_{k = 4}^{13}(5k + 8)=5\sum_{k = 4}^{13}k+\sum_{k = 4}^{13}8$.

Step2: Use sum - of - arithmetic - series formula for $\sum_{k = 4}^{13}k$

The sum of an arithmetic series $\sum_{i = 1}^{n}i=\frac{n(n + 1)}{2}$. So, $\sum_{k = 4}^{13}k=\sum_{k = 1}^{13}k-\sum_{k = 1}^{3}k$.
$\sum_{k = 1}^{13}k=\frac{13\times(13 + 1)}{2}=\frac{13\times14}{2}=91$, $\sum_{k = 1}^{3}k=\frac{3\times(3 + 1)}{2}=\frac{3\times4}{2}=6$. Then $\sum_{k = 4}^{13}k=91 - 6=85$.

Step3: Calculate $\sum_{k = 4}^{13}8$

Since we are adding 8 a total of $13-4 + 1 = 10$ times, $\sum_{k = 4}^{13}8=8\times(13 - 4+1)=8\times10 = 80$.

Step4: Calculate the original sum

$5\sum_{k = 4}^{13}k+\sum_{k = 4}^{13}8=5\times85+80=425 + 80=505$.

Answer:

505