Question

find the sum of the finite arithmetic series. \\( \sum_{n=1}^{10} (6n + 5) \\)

Explanation:

Step1: Recall the formula for the sum of an arithmetic series

The sum of the first \( n \) terms of an arithmetic series \( \sum_{k = 1}^{n} a_k \) is given by \( S_n=\frac{n}{2}(a_1 + a_n) \), where \( a_1 \) is the first term and \( a_n \) is the \( n \)-th term.

First, we need to find \( a_1 \) and \( a_{10} \) for the series \( \sum_{n = 1}^{10}(6n + 5) \).

Step2: Find the first term \( a_1 \)

When \( n = 1 \), substitute into \( a_n=6n + 5 \):
\( a_1=6(1)+5=6 + 5 = 11 \)

Step3: Find the 10th term \( a_{10} \)

When \( n = 10 \), substitute into \( a_n=6n + 5 \):
\( a_{10}=6(10)+5=60 + 5 = 65 \)

Step4: Calculate the sum \( S_{10} \)

We know that \( n = 10 \), \( a_1 = 11 \), and \( a_{10}=65 \). Using the sum formula \( S_n=\frac{n}{2}(a_1 + a_n) \):
\( S_{10}=\frac{10}{2}(11 + 65)=5\times76 = 380 \)

Answer:

\( 380 \)