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Chapter 13: Problem 9

The fourth term in an arithmetic sequence is \(-6,\) and the 10th term is 5. Find the common difference and the first term.

Short Answer

Expert verified
The common difference is \(\frac{11}{6}\) and the first term is -11.5.

Step by step solution

01

Recall the Formula for Arithmetic Sequence

In an arithmetic sequence, each term after the first is obtained by adding a constant, called the common difference, to the previous term. The general formula for the nth term is given by \( a_n = a_1 + (n-1) imes d \), where \( a_1 \) is the first term and \( d \) is the common difference.
02

Set Up Equations for Known Terms

We know that the fourth term \( a_4 = -6 \) and the tenth term \( a_{10} = 5 \). Using the formula from Step 1, we can write two equations:1. \( a_1 + 3d = -6 \)2. \( a_1 + 9d = 5 \) These equations will help us find \( a_1 \) and \( d \).
03

Subtract Equations to Solve for the Common Difference

Subtract the first equation from the second to eliminate \( a_1 \):\[(a_1 + 9d) - (a_1 + 3d) = 5 - (-6)\]Simplifying, we get:\[6d = 11\]Thus, the common difference \( d = \frac{11}{6} \approx 1.833 \).
04

Substitute the Common Difference to Find the First Term

Substitute \( d = \frac{11}{6} \) back into the first equation:\[a_1 + 3\left(\frac{11}{6}\right) = -6\]This gives us:\[a_1 + \frac{33}{6} = -6\]Simplifying:\[a_1 + 5.5 = -6\]Therefore, \( a_1 = -6 - 5.5 = -11.5 \).

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Key Concepts

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

Common Difference
In an arithmetic sequence, the common difference is a crucial component that determines how the sequence progresses from one term to the next. Simply put, the common difference is the fixed amount that you add (or subtract) to get from one term to another in the sequence.
Let's look at it more closely:
  • Imagine you have an arithmetic sequence starting at the first term, and each subsequent number is created by adding the common difference to the previous number.
  • The common difference is represented by "d" in the formula for the nth term.
  • In some sequences, the common difference can be negative, meaning each term is smaller than the one before it.
  • If you add the same number each time, you get a sequence that moves up or down in a straight line, rather than a curve.
  • Knowing the common difference helps you predict any term in the sequence without having to list out all the terms.
The step-by-step solution showed us how to find the common difference by using known terms of the sequence. By setting up equations and solving for "d," we found that the common difference is approximately 1.833.
First Term
The first term is the starting point of an arithmetic sequence. Denoted by \( a_1 \), it sets the stage for the entire sequence by providing the base term from which all other terms are calculated.
Understanding the first term is essential because:
  • It's the foundation of the sequence, and every other term is derived by applying the common difference to this starting point.
  • By knowing \( a_1 \), you can predict any specific term using the general formula.
  • The first term can be zero, positive, or negative, depending on the sequence.
In the given problem, we found the first term to be \( -11.5 \) by using the known values of other terms and the common difference. Once we have the first term and the common difference, the rest of the sequence can be easily constructed.
General Formula for nth Term
The general formula for finding the nth term of an arithmetic sequence is a powerful tool that links the first term, the common difference, and the term number together. This formula is expressed as:
  • \( a_n = a_1 + (n-1) \times d \)
Where:
  • \( a_n \) is the nth term of the sequence.
  • \( a_1 \) is the first term.
  • \( n \) is the term number.
  • \( d \) is the common difference.
This formula is beneficial because it allows you to calculate any term in the sequence without writing out all previous terms. For example:
  • If you wanted to find the 100th term, you can simply plug \( n = 100 \), your known \( a_1 \), and \( d \) into the formula.
The general formula is especially useful in solving problems where only some of the terms are known, as it provides an algebraic method for finding missing values, as shown in the original solution.

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Most popular questions from this chapter

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