Find each indicated sum. ∑k=15k(k+4)\sum_{k=1}^{5} k(k+4)

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 33

9장, 문제 33

Find each indicated sum. $\sum_{k = 1}^{5} k (k + 4)$

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Identify the sum notation: you need to find the sum of the expression $k(k+4)$ as $k$ goes from 1 to 5, which is written as $\sum_{k=1}^{5} k(k+4)$.

Expand the expression inside the summation: $k(k+4) = k^2 + 4k$.

Rewrite the sum as the sum of two separate sums: $\sum_{k=1}^{5} (k^2 + 4k) = \sum_{k=1}^{5} k^2 + \sum_{k=1}^{5} 4k$.

Use the properties of summation to factor out constants: $\sum_{k=1}^{5} 4k = 4 \sum_{k=1}^{5} k$.

Calculate each sum separately using formulas: $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$ and $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$, then substitute $n=5$ and add the results.

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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. The expression 5Σk=1 means to sum the terms as k goes from 1 to 5. Understanding this notation is essential to correctly evaluate the sum.

Evaluating Polynomial Expressions

Each term in the sum involves evaluating the polynomial k(k+4) for each integer k from 1 to 5. This requires substituting values of k into the expression and simplifying before summing the results.

Properties of Finite Sums

Finite sums can be broken down into sums of simpler expressions, such as sums of k and sums of constants. Using properties like linearity of summation helps simplify calculations by separating and summing individual parts.

Find each indicated sum. ∑k=15k(k+4)\(\sum\)_{k=1}^{5} k(k+4)