Question

8 evaluate the finite arithmetic series: 6 + 11 + 16 + \cdots + 76 $s_n = n \left(\frac{a_1 + a_n}{2}\
ight)$

Explanation:

Step1: Find the common difference \(d\)

The arithmetic sequence has \(a_1 = 6\), and the next term \(a_2 = 11\). So \(d=a_2 - a_1=11 - 6 = 5\).

Step2: Find the number of terms \(n\)

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\). We know \(a_n = 76\), \(a_1 = 6\), \(d = 5\).
Substitute into the formula: \(76=6+(n - 1)\times5\)
Subtract 6 from both sides: \(70=(n - 1)\times5\)
Divide both sides by 5: \(14=n - 1\)
Add 1 to both sides: \(n = 15\)

Step3: Calculate the sum \(S_n\)

Use the formula \(S_n=n\times\frac{a_1 + a_n}{2}\), where \(n = 15\), \(a_1 = 6\), \(a_n = 76\).
\(S_{15}=15\times\frac{6 + 76}{2}=15\times\frac{82}{2}=15\times41 = 615\)

Answer:

\(615\)