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Chapter 11: Problem 18

Find the first term and the common difference. Find \(a_{20}\) when \(a_{1}=14\) and \(d=-3\)

Short Answer

Expert verified
The 20th term is \(-43\).

Step by step solution

01

Identify the known values

Recognize the given values: the first term (\(a_1\)) and the common difference (\(d\)). Here, \(a_1 = 14\) and \(d = -3\).
02

Understand the formula for the nth term

The formula for the nth term (\(a_n\)) of an arithmetic sequence is given by \[a_n = a_1 + (n - 1)d\]
03

Substitute the known values into the formula

Substitute \(a_1 = 14\), \(d = -3\), and \(n = 20\) into the formula: \[a_{20} = 14 + (20 - 1)(-3)\]
04

Perform the calculations

Calculate the value inside the parentheses: \(20 - 1 = 19\). Then multiply by the common difference \(19 \times -3 = -57\). Finally, add this value to the first term: \[14 + (-57) = 14 - 57= -43\]
05

State the final answer

The value of the 20th term (\(a_{20}\)) is \(-43\).

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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 a crucial value because it's the starting point of the entire series. Denoted by \(a_1\), it sets the stage for the subsequent terms. For instance, in the given exercise, the first term is \(a_1 = 14\). This means the sequence starts at 14 and evolves according to the common difference.Why is the first term important?
  • It defines the initial position of the series.
  • All other terms in the sequence are built upon this primary value.
It's like the seed from which the sequence grows. Once you know \(a_1\), you can use it to determine any other term in the sequence using the nth term formula.
common difference
The common difference, represented by \(d\), is the consistent amount that separates successive terms in an arithmetic sequence. It indicates how much you add or subtract to move from one term to the next. For the given exercise, the common difference is \(d = -3\).Characteristics of the common difference:
  • If \(d\) is positive, the sequence increases.
  • If \(d\) is negative, the sequence decreases.
  • If \(d\) is zero, all terms in the sequence are the same.
In our example, since \(d\) is -3, this tells us that each term is 3 less than the previous one. This is vital to understanding how the sequence progresses or regresses.
nth term formula
To find any term in an arithmetic sequence, you use the nth term formula. This formula helps you determine the value of the sequence at any position \(n\). The formula is:\[a_n = a_1 + (n - 1)d\]Let's break it down:
  • \(a_n\): the nth term you want to find
  • \(a_1\): the first term of the sequence
  • \(n\): the position of the term in the sequence
  • \(d\): the common difference
Example from the exercise:
We need to find the 20th term (\

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