Problem 22 - College Algebra

Question

In Exercises \(15-22,\) find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d\) Find \(a_{70}\) when \(a_{1}=-32, d=4\)

Step-by-Step Solution

Answer

The 70th term of the arithmetic sequence (a_{70}) is 244.

Step 1: Insert the given values into the formula

Insert the given values: \(a_{1} = -32\), \(d = 4\), and \(n = 70\) into the formula \(a_{n} = a_{1} + (n-1)d\). This results in \(a_{70} = -32 + (70-1)4\)

Step 2: Simplify the expression

Calculate the expression in the parentheses first, according to the Order of Operations (BIDMAS/BODMAS). The calculation becomes \(a_{70} = -32 + 69*4\).

Step 3: Solve for \(a_{70}\)

Multiply and add to solve the equation. The result is \(a_{70} = -32 + 276 = 244\)

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers with a specific pattern: each number is obtained by adding a fixed value to the previous one. This consistent interval is known as the 'common difference'. In simpler terms, if you start with a number and keep adding (or subtracting, if the common difference is negative) the same value, you've created an arithmetic sequence. For example, if we begin with 3 and have a common difference of 5, the sequence looks like this: 3, 8, 13, 18, and so on. Each number is the result of adding 5 to the previous number. It's like climbing steps where each step is of equal height - that height being the common difference.

The beauty of arithmetic sequences is in their predictability. Once you know the first term and the common difference, you can figure out any term in the sequence without having to write out the whole thing. This characteristic makes arithmetic sequences particularly handy in various fields, such as finance for calculating interest, or computer science for creating loops.

Common Difference

The 'common difference' in an arithmetic sequence is the key to understanding its behavior. It is the consistent difference between consecutive terms and it can be any real number: positive, negative, or even zero. In our exercise, the common difference is 4. This means that every term is 4 more than the one before it.

If we visualize this on a number line, each term is a fixed step to the right of the last one if the common difference is positive. If it is negative, each step goes to the left. A zero common difference would mean all terms are the same, as we're not really moving away from the initial number. Understanding the common difference helps us predict the progression of the sequence and calculate any term within it. It's much like knowing that every step you take is exactly one meter - you can easily calculate how far you'll be after 70 steps.

Sequence Term Formula

When you need to find a specific term in an arithmetic sequence, you use the 'sequence term formula': \[a_n = a_1 + (n - 1)d\]. In this formula, \(a_n\) represents the nth term you're trying to find, \(a_1\) is the first term, \(n\) stands for the term's position in the sequence, and \(d\) is the common difference.Let's break it down using the exercise as an example. The sequence starts at \(-32\) with a common difference of 4, and we're looking for the 70th term. By substituting into the formula, the calculation to find \(a_{70}\) is straightforward. You multiply the common difference by 69 (because we're looking at the interval between 70 terms, which is actually 69 steps), then add this result to the first term.This formula is powerful because it lets you leap straight to any term without calculating all the ones in between. Imagine if you wanted to know where you'd land on step 1000 without taking every single step before it - the formula is like a shortcut that instantly transports you there.

The sequence term formula is a core concept not only for solving textbook exercises, but also for real-world applications such as planning savings or understanding repetitive patterns. It encapsulates the essence of arithmetic sequences and their predictable nature.