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Chapter 11: Problem 74

Find the first term of the arithmetic sequence with a common difference of \(-5\) if its 23 rd term is \(-625\).

Short Answer

Expert verified
The first term of the sequence is \(-515\).

Step by step solution

01

Understand the Problem

We need to find the first term of an arithmetic sequence. We are given two pieces of information: the common difference, which is \(-5\), and the 23rd term, which is \(-625\).
02

Recall the Arithmetic Sequence Formula

The formula for finding the n-th term of an arithmetic sequence is:\[ a_n = a_1 + (n-1) imes d \]where \(a_n\) is the n-th term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.
03

Plug in the Known Values

We know \(a_{23} = -625\), the common difference \(d = -5\), and the term number \(n = 23\). Substitute these values into the sequence formula:\[ -625 = a_1 + (23-1) imes (-5) \]
04

Simplify the Equation

First, simplify \(23 - 1\) to get 22. Substitute into the equation:\[ -625 = a_1 + 22 imes (-5) \]Calculate \(22 imes (-5) = -110\), so the equation becomes:\[ -625 = a_1 - 110 \]
05

Solve for the First Term

Add \(110\) to both sides of the equation to isolate \(a_1\):\[ -625 + 110 = a_1 \]Simplify the left side:\[ a_1 = -515 \]

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Key Concepts

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

Common Difference
Arithmetic sequences are a type of number pattern where the difference between consecutive terms is consistent. This consistent difference is known as the *common difference*. In our example, the common difference is \(-5\).
  • The fact that it is negative means that each term in the sequence is decreasing by 5 units from the previous term.
  • This negative change signifies that each subsequent term is smaller than the prior one.
  • In arithmetic sequences, the common difference \(d\) plays a pivotal role in determining the nature and direction of the sequence.
Understanding the common difference helps in predicting and formulating any term within the sequence, making it a crucial component of arithmetic progressions.
n-th Term Formula
To pinpoint a specific term in an arithmetic sequence, we use the n-th term formula. This formula is handy for finding any term when we know: the first term (\(a_1\)), the common difference (\(d\)), and the position of the term (\(n\)).
  • The n-th term formula is given by: \(a_n = a_1 + (n-1) \times d\).
  • This equation accounts for the number of times the common difference is added (or subtracted) to reach the n-th term.
  • Using this, we can calculate the value of any term without needing to list the entire sequence.
In solving our specific problem, we applied the formula to find the first term given the 23rd term and the common difference, emphasizing the formula's versatility.
First Term
The first term of an arithmetic sequence (\(a_1\)) is foundational because it serves as the starting point of the sequence. In many problems, finding the first term is crucial for unlocking the nature of the sequence.
  • In our case, the problem asked us to find \(a_1\) when given the 23rd term \(-625\) and the common difference \(-5\).
  • We rearranged the n-th term formula to solve for \(a_1\): \( \ a_1 = a_n - (n-1) \times d \).
  • Upon calculation, this provided us with a value of \(-515\), showcasing the utility of arithmetic principles in determining initial values.
Understanding the first term helps us reconstruct the entire sequence, providing a clear picture of its structure and behavior.

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Most popular questions from this chapter

Write the summation notation for \(3+4+5+6\) without using \(k=1\) in your answer.

Is there a geometric mean between \(-5\) and \(5 ?\)

What is the difference between a sequence and a series?

Find each sum. See Example 9. $$ \sum_{k=2}^{4}\left(-k^{2}\right) $$

Write the first four terms of the geometric sequence if its first term is \(-64\) and its sixth term is \(-2\)
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