Find each indicated sum. ∑i=165i\sum_{i=1}^{6} 5i

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 29

9장, 문제 29

Find each indicated sum. $\sum_{i = 1}^{6} 5 i$

검증된 단계별 안내

Identify the summation notation given: $\sum_{i=1}^{6} 5i$. This means you need to find the sum of the expression \$5i$ as $i$ goes from 1 to 6.

Rewrite the summation by expanding the terms: $5(1) + 5(2) + 5(3) + 5(4) + 5(5) + 5(6)$.

Factor out the constant 5 from the summation: $5 \sum_{i=1}^{6} i$.

Use the formula for the sum of the first $n$ natural numbers: $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$. Here, $n=6$.

Substitute $n=6$ into the formula and multiply by 5: $5 \times \frac{6 \times (6+1)}{2}$. This expression represents the sum you need to calculate.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Summation Notation (Sigma Notation)

Summation notation uses the Greek letter sigma (Σ) to represent the sum of a sequence of terms. It specifies the index of summation, the lower and upper limits, and the expression to be summed. For example, Σ from i=1 to 6 of 5i means adding 5 times each integer from 1 to 6.

Arithmetic Series

An arithmetic series is the sum of terms in an arithmetic sequence, where each term increases by a constant difference. In this problem, 5i forms an arithmetic sequence with a common difference of 5. Understanding this helps in applying formulas to find the sum efficiently.

Formula for the Sum of an Arithmetic Series

The sum of the first n terms of an arithmetic series can be found using the formula S_n = n/2 (first term + last term). This formula simplifies the calculation by avoiding term-by-term addition, making it useful for quickly finding sums like Σ 5i from i=1 to 6.

Find each indicated sum. ∑i=165i\(\sum\)_{i=1}^{6} 5i