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Magazine Chapter 11: Problem 91
For the following exercises, find the first term given two terms from an arithmetic sequence. Find the first term or \(a_{1}\) of an arithmetic sequence if \(a_{7}=21\) and \(a_{15}=42\)
Short Answer
Expert verified
The first term \(a_1\) is 5.25.
Step by step solution
01
Understand the Problem
We are given two terms of an arithmetic sequence: the 7th term \(a_7 = 21\) and the 15th term \(a_{15} = 42\). The goal is to find the first term \(a_1\). To achieve this, we'll use the formula for the \(n\)-th term of an arithmetic sequence, \(a_n = a_1 + (n-1)d\), where \(d\) is the common difference.
02
Set up Equations from Given Terms
Using the formula, \(a_7 = a_1 + 6d = 21\) and \(a_{15} = a_1 + 14d = 42\). We now have two equations:1. \(a_1 + 6d = 21\)2. \(a_1 + 14d = 42\).
03
Eliminate \(a_1\) to Find \(d\)
Subtract the first equation from the second:\[(a_1 + 14d) - (a_1 + 6d) = 42 - 21\]Simplifying gives:\[8d = 21\]Solve for \(d\) to find:\[d = \frac{21}{8} = 2.625\].
04
Substitute \(d\) to Find \(a_1\)
Substitute \(d = 2.625\) back into one of the original equations (for instance, \(a_1 + 6d = 21\)):\[a_1 + 6(2.625) = 21\]Calculate:\[a_1 + 15.75 = 21\]Solve for \(a_1\):\[a_1 = 21 - 15.75 = 5.25\].
05
Verify the Solution
Check the values by calculating the 15th term using \(a_1 = 5.25\) and \(d = 2.625\):\[a_{15} = 5.25 + 14(2.625) = 5.25 + 36.75 = 42\].Since the calculation matches the given \(a_{15} = 42\), the solution is verified.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sequence Formula
An arithmetic sequence is a list of numbers with a constant difference, known as the common difference, between consecutive terms. It can be represented by using a specific formula. This formula helps you find any term in the sequence, which is especially useful if only a few terms are known.The general formula for finding the nth term in an arithmetic sequence is:
- \(a_n = a_1 + (n-1)d\)
- \(a_n\) is the nth term you're trying to find,
- \(a_1\) is the first term of the sequence, and
- \(d\) is the common difference between terms.
Common Difference
The common difference in an arithmetic sequence is the constant amount each term increases or decreases from the previous term. It's a crucial component of the formula for arithmetic sequences.To find the common difference \(d\), consider two terms in the sequence, such as \(a_7\) and \(a_{15}\). Using their positions and values, you can set up an equation that isolates \(d\). For example, if:
- \(a_7 = 21\)
- \(a_{15} = 42\)
- \(a_7 = a_1 + 6d = 21\)
- \(a_{15} = a_1 + 14d = 42\)
- \(8d = 21\)
- \(d = \frac{21}{8} = 2.625\)
Term Calculation
Calculating a specific term in an arithmetic sequence involves using the sequence formula with known values of \(a_1\) and \(d\). For instance, we wanted to find \(a_1\) in the given exercise.Once the common difference \(d = 2.625\) was known, you simply need to resubstitute this back into one of the earlier equations like \(a_1 + 6d = 21\). This allows you to calculate \(a_1\):
- \(a_1 + 6(2.625) = 21\)
- \(a_1 + 15.75 = 21\)
- \(a_1 = 21 - 15.75 = 5.25\)
- \(a_n = a_1 + (n-1)d\)
Algebraic Equations
Solving problems like these often involves setting up and solving algebraic equations. Given multiple terms from a sequence, you can create equations based on the formula for the nth term.In this exercise, you have two equations from the terms \(a_7\) and \(a_{15}\). It looks like this:
- \(a_1 + 6d = 21\)
- \(a_1 + 14d = 42\)
- \((a_1 + 14d) - (a_1 + 6d) = 42 - 21\)
- \(8d = 21\)
