Problem 34 - College Algebra

Question

Find each indicated sum. $$\sum_{i=1}^{2}(k-3)(k+2)$$

Step-by-Step Solution

Answer

The sum of the series \(\sum_{i=1}^{2}(k-3)(k+2)\) is -10.

Step 1: Substitute \(k=1\) into the expression

Substitute \(k=1\) into the series expression \((k-3)(k+2)\) to get \((1-3)(1+2) = -2*3 = -6\)

Step 2: Substitute \(k=2\) into the expression

Substitute \(k=2\) into the series expression \((k-3)(k+2)\) to get \((2-3)(2+2) = -1*4 = -4\)

Step 3: Sum the results

Now we sum the results from steps 1 and 2. The sum \(-6 + -4 = -10\)

Key Concepts

Series and Sequence

Let's dive into the fascinating world of series and sequences, a critical topic in mathematics that concerns an ordered list of numbers. Generally, a sequence is like a list of beads on a string, one after another, often following a specific rule to determine which bead comes next. A series, on the other hand, is the sum of the terms of a sequence. Imagine if you decided to glue all those beads together to form a single object, the series would be the total length of your beaded creation.

In the context of our problem, we are looking at a finite series which is just a series with a start and an end point, no infinite tail here! The notation \(\sum\) represents the sum of a series, with \(i=1\) showing where we start summing, and \(2\) is where we stop. In the exercise given, we are summing up the values of the algebraic expression \( (k-3)(k+2) \) when \(k\) takes on the values 1 and 2.

Understanding how to work with series is crucial, not just in pure math but also in fields like physics and economics, where it's used to model phenomena over time or to calculate accumulated interest, respectively.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables (like our friend \(k\)) and arithmetic operations such as addition, subtraction, multiplication, and division. It's like a recipe that tells you how to mix these ingredients together. They don't usually contain an equals sign, that's more of an equation's thing.

In our particular algebraic expression, \( (k-3)(k+2) \), we spot a pair of binomials being multiplied together. Binomials are like sandwiches with just two ingredients, and when they come together, we get a new recipe with potentially four terms after following the FOIL method—First, Outside, Inside, Last. However, our expression all boils down to two straightforward operations: we subtract 3 from \(k\) and add 2 to \(k\), then multiply these two outcomes together.

Grasping algebraic expressions allows students to tackle a range of problems from simple to complex, and apply these skills in countless scenarios, making it a bread-and-butter topic in algebra.

Substitution Method

Substitution is a super handy tool, like a 'swap-and-solve' technique. It simply means replacing a variable with its value or another expression. Imagine having a password-protected note, and every time you see the word 'password,' you swap it with the actual password to understand the message. That's substitution!

In the steps of our solution, the substitution came into play when we replaced \(k\) with its given values of 1 and 2. We first calculate what happens when \(k\) is 1 and then do it all over again when \(k\) is 2. Each substitution simplifies the algebraic expression down to a number, making it possible to find the final sum required in the problem.

Removing the ambiguity of variables using substitution is a critical skill across various areas of mathematics, from solving simple equations to unraveling more complex functions and calculus problems.