In Exercises 29–42, find each indicated sum. 5Σk=1 k(k+4)

Verified step by step guidance

1. Identify the given summation: $\sum _{k=1}^{5}k(k+4)$.
2. Recognize that this is a finite sum where $k$ ranges from 1 to 5.
3. For each value of $k$, calculate the expression $k(k+4)$.
4. List the results of the expression for each $k$: $1(1+4),2(2+4),3(3+4),4(4+4),5(5+4)$.
5. Add the results from the previous step to find the total sum.

Key Concepts

Summation Notation

Summation notation, represented by the Greek letter sigma (Σ), is a concise way to express the sum of a sequence of terms. In the expression 5Σk=1 k(k+4), it indicates that we need to sum the values of the function k(k+4) for each integer k from 1 to 5. Understanding how to interpret and manipulate summation notation is essential for solving problems involving series.

Polynomial Functions

The expression k(k+4) is a polynomial function of k, specifically a quadratic function. It can be expanded to k^2 + 4k, which helps in calculating the sum more easily. Recognizing the structure of polynomial functions is important for simplifying expressions and performing operations like summation.

Evaluating Sums

To evaluate the sum 5Σk=1 k(k+4), one must compute the value of the polynomial for each integer k from 1 to 5 and then add those results together. This process involves substituting values into the polynomial, calculating each term, and then performing the final addition. Mastery of this technique is crucial for solving summation problems in algebra.