Problem 9
Question
Write the first five terms of each arithmetic sequence. Do not use a calculator. $$a_{1}=5, d=-2$$
Step-by-Step Solution
Verified Answer
The first five terms are 5, 3, 1, -1, -3.
1Step 1: Understand the problem
We are asked to find the first five terms of an arithmetic sequence, which begins with the first term \(a_1 = 5\) and has a common difference \(d = -2\). An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.
2Step 2: Recall the formula for arithmetic sequences
In an arithmetic sequence, each term can be calculated using the formula \(a_n = a_1 + (n-1) imes d\), where \(a_n\) is the \(n\)-th term, \(a_1\) is the first term, and \(d\) is the common difference.
3Step 3: Calculate the first term
Use the formula to verify the first term: \(a_1 = 5\). The first term of the sequence is already given, so \(a_1 = 5\).
4Step 4: Calculate the second term
Using the formula \(a_2 = a_1 + (2-1) imes d\), we substitute \(a_1 = 5\) and \(d = -2\): \(a_2 = 5 + (1) imes (-2) = 5 - 2 = 3\). The second term is \(3\).
5Step 5: Calculate the third term
Using the formula \(a_3 = a_1 + (3-1) imes d\), we substitute \(a_1 = 5\) and \(d = -2\): \(a_3 = 5 + 2 imes (-2) = 5 - 4 = 1\). The third term is \(1\).
6Step 6: Calculate the fourth term
Using the formula \(a_4 = a_1 + (4-1) imes d\), we substitute \(a_1 = 5\) and \(d = -2\): \(a_4 = 5 + 3 imes (-2) = 5 - 6 = -1\). The fourth term is \(-1\).
7Step 7: Calculate the fifth term
Using the formula \(a_5 = a_1 + (5-1) imes d\), we substitute \(a_1 = 5\) and \(d = -2\): \(a_5 = 5 + 4 imes (-2) = 5 - 8 = -3\). The fifth term is \(-3\).
8Step 8: List the first five terms
The first five terms of the arithmetic sequence are obtained from each of the calculations: 5, 3, 1, -1, -3.
Key Concepts
Common DifferenceSequence FormulaTerm Calculation
Common Difference
In an arithmetic sequence, understanding the common difference is crucial. The common difference, denoted by the letter "\(d\)", indicates the fixed amount by which each term in the sequence increases or decreases as you move from one term to the next. In our exercise, the common difference \(d\) is \(-2\).
This means every following term is obtained by subtracting 2 from the previous term. To calculate subsequent terms effectively, always keep the common difference in mind:
This means every following term is obtained by subtracting 2 from the previous term. To calculate subsequent terms effectively, always keep the common difference in mind:
- If the common difference is positive, the sequence will increase.
- If it is negative, the sequence decreases.
- A common difference of zero would mean that each term in the sequence is the same.
Sequence Formula
The sequence formula is a powerful tool that allows us to find any term in an arithmetic sequence. This formula is expressed as \(a_n = a_1 + (n-1) \times d\). Let's break this formula down:
\(a_3 = 5 + (3-1) \times (-2) = 5 + 2 \times (-2) = 5 - 4 = 1.\)
Using this formula, you can predict any term in the sequence without having to list all the previous terms, which is especially handy with larger sequences.
- \(a_n\) represents the \(n\)-th term of the sequence.
- \(a_1\) is the first term of the sequence.
- \(n\) is the term number you are trying to find.
- \(d\) is the common difference.
\(a_3 = 5 + (3-1) \times (-2) = 5 + 2 \times (-2) = 5 - 4 = 1.\)
Using this formula, you can predict any term in the sequence without having to list all the previous terms, which is especially handy with larger sequences.
Term Calculation
In arithmetic sequences, calculating each term after the first is a systematic process. Once you know the first term and the common difference, you can calculate any subsequent term directly. Here's how you can approach it:
Start with the formula \(a_n = a_1 + (n-1) \times d\), which we've already discussed. For each term you need to find:
\(a_2 = 5 + (2-1) \times (-2) = 3.\)
Continue this process for the 3rd, 4th, and 5th terms to achieve:
3rd term: \(1\), 4th term: \(-1\), and 5th term: \(-3\).
Mastering this process ensures accuracy and efficiency when dealing with arithmetic sequences in any context.
Start with the formula \(a_n = a_1 + (n-1) \times d\), which we've already discussed. For each term you need to find:
- Substitute the corresponding \(n\) value for the term's position.
- Plug in the known \(a_1\) and \(d\) values.
- Solve to find \(a_n\).
\(a_2 = 5 + (2-1) \times (-2) = 3.\)
Continue this process for the 3rd, 4th, and 5th terms to achieve:
3rd term: \(1\), 4th term: \(-1\), and 5th term: \(-3\).
Mastering this process ensures accuracy and efficiency when dealing with arithmetic sequences in any context.
Other exercises in this chapter
Problem 8
Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=(-2)^{n}(n)$$
View solution Problem 9
Use mathematical induction to prove each statement. Assume that \(n\) is a positive integer. $$3+3^{2}+3^{3}+\cdots+3^{n}=\frac{3\left(3^{n}-1\right)}{2}$$
View solution Problem 9
Evaluate each expression. Do not use a calculator. $$3 ! \cdot 4$$
View solution Problem 9
CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$a_{4}=243, r=-3$$
View solution