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Chapter 11: Problem 26

Find the first term and the common difference. Find \(a_{1}\) when \(d=8\) and \(a_{11}=26\)

Short Answer

Expert verified
The first term, a_{1}, is -54.

Step by step solution

01

Understand the Formula for Arithmetic Sequences

An arithmetic sequence has the form: a_{n} = a_{1} + (n-1)da_{n} is the n-th term, a_{1} is the first term, n is the term number, and d is the common difference.
02

Set Up the Equation for the Given Term

Use the given term information: a_{11} = 26. Substitute the given values into the arithmetic sequence formula: a_{11} = a_{1} + (11-1)d.
03

Substitute Known Values

We know that a_{11} = 26 and d = 8. Substitute these values into the equation: 26 = a_{1} + 10(8).
04

Solve for a_{1}

Simplify the equation: 26 = a_{1} + 80Subtract 80 from both sides: a_{1} = 26 - 80a_{1} = -54.

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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 represented by \(a_1\). This term is crucial as it sets the starting point for the sequence. To find the first term, you often need some additional information like one of the sequence values and the common difference.

For example, in the given problem, we need to find the first term \(a_1\) when the 11th term \(a_{11}\) is 26, and the common difference \(d\) is 8. We use these pieces of information in the arithmetic sequence formula to solve for \(a_1\), which is found to be -54.
Common Difference
The common difference, denoted as \(d\), is a key characteristic of an arithmetic sequence. It represents the constant amount added to each term to get to the next term in the sequence.

To solve problems involving arithmetic sequences, knowing the common difference is essential because it helps define the relationship between consecutive terms. In the provided exercise, the common difference \(d\) is given as 8. This means each term is 8 more than the previous term.
Arithmetic Sequence Formula
The fundamental formula for calculating any term in an arithmetic sequence is: \[ a_n = a_1 + (n-1)d \]

Here, \(a_n\) is the n-th term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.

This formula allows us to calculate any term in the sequence when we know the first term and the common difference. In our problem, the formula helps to derive the first term \(a_1\) given specific values for \(a_{11}\) and \(d\). Implementing the known values into the formula, we set up the equation to solve for \(a_1\).
Term Number
The term number, represented as \(n\), indicates the position of a term within an arithmetic sequence. This number helps us identify which term we are referring to or need to calculate.

When determining the value of a specific term, knowing its position (n-th term) is necessary. For example, in the exercise, we deal with the 11th term (\(a_{11}\)) where \(n=11\). By inserting this term number into the arithmetic sequence formula, we can find the value of that term if we know the first term and the common difference.

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Most popular questions from this chapter

Carrie saves money in an arithmetic sequence: \(\$ 700\) for the first year, another \(\$ 850\) the second, and so on, for 20 years. How much does she save in all (disregarding interest)?

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