Question

Evaluate the series LaTeX: \sum_{n=3}^{8}{6n}.

Answer 1

To evaluate the series \(\sum_{n=3}^{8}{6n}\), we first recognize that we can factor out the constant \(6\) from the summation:

\[
\sum_{n=3}^{8} 6n = 6 \sum_{n=3}^{8} n
\]

Next, we need to compute the sum \(\sum_{n=3}^{8} n\). This sum consists of the integers from \(3\) to \(8\):

\[
\sum_{n=3}^{8} n = 3 + 4 + 5 + 6 + 7 + 8
\]

Calculating this step by step, we have:

- \(3 + 4 = 7\)
- \(7 + 5 = 12\)
- \(12 + 6 = 18\)
- \(18 + 7 = 25\)
- \(25 + 8 = 33\)

Thus, we have:

\[
\sum_{n=3}^{8} n = 33
\]

Now, substituting back into our earlier expression, we get:

\[
6 \sum_{n=3}^{8} n = 6 \times 33 = 198
\]

Therefore, the value of the series \(\sum_{n=3}^{8} 6n\) is

\[
\boxed{198}
\]