Question

find the sum of the finite arithmetic series.
\\(\sum_{n = 0}^{17} (6n + 3)\\)

Explanation:

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

The sum \( S_n \) of the first \( n \) terms of an arithmetic series is given by \( S_n=\frac{n}{2}(a_1 + a_n) \), where \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the \( n \)-th term. Also, for an arithmetic sequence with \( a_n=a_1+(n - 1)d \), where \( d \) is the common difference.

First, we need to find the number of terms, the first term, and the last term.

The series is \( \sum_{n = 0}^{17}(6n+3) \). When \( n = 0 \), \( n=1,\cdots,n = 17 \). The number of terms \( N=17 - 0+1=18 \) terms.

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

When \( n = 0 \), \( a_1=6(0)+3 = 3 \).

Step3: Find the last term (\( a_{18} \)) (since there are 18 terms, \( n \) from 0 to 17, so the 18 - th term corresponds to \( n = 17 \))

When \( n = 17 \), \( a_{18}=6(17)+3=102 + 3=105 \).

Step4: Calculate the sum

Using the sum formula \( S_N=\frac{N}{2}(a_1 + a_N) \), where \( N = 18 \), \( a_1=3 \), \( a_N = 105 \).

\( S_{18}=\frac{18}{2}(3 + 105)=9\times108 = 972 \)

Answer:

972