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Chapter 11: Problem 19

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 is 5 and the common difference is 4.

Step by step solution

01

Identify the Given Information

The problem provides the common difference ( d=4 ) and the 8th term ( a_{8}=33 ).
02

Use the Arithmetic Sequence Formula

The general term of an arithmetic sequence is given by a_{n} = a_{1} + (n-1)d .
03

Substitute Known Values

Substitute a_{8} for a_{n} , 8 for n , and 4 for d in the arithmetic sequence formula. 33 = a_{1} + (8-1) * 4 .
04

Simplify the Equation

Simplify the equation: 33 = a_{1} + 7 * 4 .
05

Solve for the First Term

Solve for a_{1} : 33 = a_{1} + 28 . Subtract 28 from both sides: 33 - 28 = a_{1} , so a_{1} = 5 .

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Key Concepts

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

common difference
The common difference is a key characteristic of an arithmetic sequence.
It is the constant amount between consecutive terms in the sequence.
This makes an arithmetic sequence linear, with each term increasing or decreasing by the same amount each time.
For instance, if the common difference is 4, each subsequent term is 4 units ahead of the previous term.
In our exercise, the common difference is 4, meaning if we start from the first term, each following term is obtained by adding 4.
This helps us predict any term when the first term and common difference are known.
first term
The first term of an arithmetic sequence is crucial because it sets the starting point for the entire sequence.
In our exercise, we had to find this first term using the 8th term and the common difference.
Knowing the first term (\(a_1\)) enables us to calculate any other term in the sequence.
For example, by knowing the first term and common difference, we can construct a formula to find any term in the sequence.
sequence formula
The sequence formula for an arithmetic sequence is a powerful tool.
This formula is \( a_{n} = a_{1} + (n-1)d \), where \(a_{n}\) represents any term in the sequence, \(a_{1}\) is the first term, \(n\) is the term number, and \(d\) is the common difference.
This formula helps you find the value of any term when you know the first term and the common difference.
In our exercise, we used this formula to find the first term by substituting the given values into it.
By switching the places of the known and unknown, you can solve for unknowns.
solving equations
Solving equations is a fundamental part of working with arithmetic sequences.
In our exercise, we turned the problem into an equation, which allowed us to find the first term.
By substituting the known values into the sequence formula, \( a_{n} \) became 33, \( n \) became 8, and \( d \) was 4.
The resulting equation was \( 33 = a_{1} + 7 \times 4 \).
Simplifying it, we solved it step-by-step to find \( a_{1} \).
This process teaches the importance of equation manipulation in finding unknown values in sequences.

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