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Chapter 11: Problem 25

Find the first term and the common difference. Find \(a_{1}\) when \(d=4\) and \(a_{8}=33\)

Short Answer

Expert verified
The first term \(a_{1}\) is 5.

Step by step solution

01

Identify the Given Information

Given the common difference \(d=4\) and the 8th term \(a_{8}=33\).
02

Use the Formula for the nth Term

The formula for the nth term of an arithmetic sequence is \(a_{n} = a_{1} + (n-1)d\). For this problem, substitute \(n=8\), \(a_{8}=33\), and \(d=4\) into the formula.
03

Substitute the Values into the Formula

Using \(a_{n} = a_{1} + (n-1)d\), substitute \(n=8\), \(a_{8}=33\), and \(d=4\): \[ 33 = a_{1} + (8-1) * 4 \]
04

Simplify and Solve for \(a_{1}\)

First, simplify the equation: \[ 33 = a_{1} + 7 * 4 \] \[ 33 = a_{1} + 28 \] Subtract 28 from both sides to solve for \(a_{1}\): \[ 33 - 28 = a_{1} \] \[ a_{1} = 5 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

first term
In an arithmetic sequence, the first term is critical because it sets the stage for the entire sequence. The first term is usually denoted as \(a_1\) and is the starting point of the sequence. For the given problem, we needed to find the first term. Given the common difference \(d\) and the 8th term \(a_8\), we used the nth term formula to solve for \(a_1\). By simplifying the equation \(33 = a_1 + 28\), we determined that the first term is \(a_1 = 5\). Understanding the first term helps us predict all subsequent terms in the sequence.
common difference
The common difference in an arithmetic sequence is the constant amount by which each term increases or decreases. It is denoted by \(d\) and remains the same throughout the sequence. Given in the problem, the common difference is \(d=4\). This means that each term is 4 units greater than the previous term. For example, if the first term is 5, the second term will be 9 (5 + 4), the third term will be 13 (9 + 4), and so on. Knowing the common difference helps us understand the pattern and behavior of the sequence.
nth term formula
The nth term formula of an arithmetic sequence gives a method to find any term in the sequence. It is represented by the equation:\[ a_n = a_1 + (n-1)d \] Here, \(a_n\) is the nth term, \(a_1\) is the first term, \(n\) is the position of the term in the sequence, and \(d\) is the common difference. Using this formula, we can find any term if we know the first term and the common difference. For example, in this problem, we used it to find \(a_8\). By substituting the given values, we verified that \(a_8 = 33\) indeed fits into the sequence, which starts with 5 and has a common difference of 4.

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