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

Verified step by step guidance

1. 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 \(\sum_{k=1}^{5} k(k+4)\).
2. Expand the expression inside the summation: \(k(k+4) = k^2 + 4k\).
3. Rewrite the sum as the sum of two separate sums: \(\sum_{k=1}^{5} (k^2 + 4k) = \sum_{k=1}^{5} k^2 + \sum_{k=1}^{5} 4k\).
4. Use the properties of summation to factor out constants: \(\sum_{k=1}^{5} 4k = 4 \sum_{k=1}^{5} k\).
5. 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.

Key Concepts

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.