Question

find the sum of the finite arithmetic series.
\\(\displaystyle\sum_{n=1}^{16} (5n - 3)\\)

Explanation:

Step1: Find the first term \(a_1\)

When \(n = 1\), substitute into \(5n - 3\):
\(a_1 = 5(1) - 3 = 2\)

Step2: Find the 16th term \(a_{16}\)

When \(n = 16\), substitute into \(5n - 3\):
\(a_{16} = 5(16) - 3 = 80 - 3 = 77\)

Step3: Use the arithmetic series sum formula \(S_n=\frac{n(a_1 + a_n)}{2}\)

Here, \(n = 16\), \(a_1 = 2\), \(a_{16} = 77\).
Substitute into the formula:
\(S_{16}=\frac{16(2 + 77)}{2}\)

Step4: Simplify the expression

First, calculate inside the parentheses: \(2 + 77 = 79\).
Then, multiply by 16: \(16\times79 = 1264\).
Finally, divide by 2: \(\frac{1264}{2} = 632\)

Answer:

\(632\)