Question

final exam
100 points possible answered: 11/22
question 12
find the sum: $-8 + -1 + 6 + \dots + (-15 + 7n)$
answer:
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Explanation:

Step1: Identify the sequence type

The sequence is \(a_k=-15 + 7k\) (let \(k\) start from 1). Check if it's arithmetic: common difference \(d=a_{k + 1}-a_k=(-15 + 7(k + 1))-(-15 + 7k)=7\). So it's an arithmetic sequence with first term \(a_1=-8\) (when \(k = 1\), \(-15+7(1)=-8\)) and last term \(a_n=-15 + 7n\).

Step2: Use arithmetic series sum formula

The sum of an arithmetic series is \(S_n=\frac{n(a_1 + a_n)}{2}\). Substitute \(a_1=-8\) and \(a_n=-15 + 7n\) into the formula:

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Answer:

\(\frac{7n^2 - 23n}{2}\)