Question

find the sum of the finite arithmetic series. \\(\sum_{n=0}^{35} (5n + 2)\\)

Explanation:

Step1: Determine the number of terms

The series is from \( n = 0 \) to \( n = 35 \), so the number of terms \( N=35 - 0+ 1=36 \).

Step2: Find the first term (\(a_1\))

When \( n = 0 \), \( a_1=5(0)+2 = 2 \).

Step3: Find the last term (\(a_N\))

When \( n = 35 \), \( a_{36}=5(35)+2=175 + 2=177 \).

Step4: Use the arithmetic series sum formula

The sum of an arithmetic series is given by \( S_N=\frac{N(a_1 + a_N)}{2} \).
Substitute \( N = 36 \), \( a_1=2 \), and \( a_{36}=177 \) into the formula:
\( S_{36}=\frac{36\times(2 + 177)}{2} \)
First, calculate the sum inside the parentheses: \( 2+177 = 179 \)
Then, \( \frac{36\times179}{2}=18\times179 \)
Calculate \( 18\times179 = 3222 \)

Answer:

\( 3222 \)