Question

question 24
find the sum
\\(\sum_{n=81}^{140} 3n =\\)
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Explanation:

Step1: Identify the series type

The series \(\sum_{n = 81}^{140} 3n\) is an arithmetic series. The first term \(a_1\) when \(n = 81\) is \(a_1=3\times81 = 243\), the last term \(a_n\) when \(n = 140\) is \(a_n=3\times140=420\). The number of terms \(k=140 - 81+1=60\).

Step2: Use arithmetic series sum formula

The sum of an arithmetic series is \(S_k=\frac{k(a_1 + a_n)}{2}\). Substitute \(k = 60\), \(a_1=243\), \(a_n = 420\) into the formula:
\(S_{60}=\frac{60\times(243 + 420)}{2}\)
First, calculate the sum inside the parentheses: \(243+420 = 663\)
Then, \(60\div2=30\), so \(30\times663 = 19890\)

Answer:

\(19890\)