Problem 25
Question
Find the \(n\)th term of a sequence whose first several terms are given. \(1,4,7,10, \dots\)
Step-by-Step Solution
Verified Answer
The nth term is \(a_n = 3n - 2\).
1Step 1: Identify the Pattern
The given sequence is 1, 4, 7, 10. Each term increases by 3 from the previous term. This sequence is arithmetic with a common difference of 3.
2Step 2: Use the Arithmetic Sequence Formula
The formula for the nth term of an arithmetic sequence is \(a_n = a_1 + (n-1) imes d\), where \(a_1\) is the first term and \(d\) is the common difference. Here \(a_1 = 1\) and \(d = 3\).
3Step 3: Substitute Known Values
Substitute \(a_1 = 1\) and \(d = 3\) into the formula: \[a_n = 1 + (n-1) imes 3\]
4Step 4: Simplify the Expression
Simplify the equation \(a_n = 1 + (n-1) imes 3\):\[a_n = 1 + 3n - 3\]\[a_n = 3n - 2\]
5Step 5: Write the Final Expression
The nth term of the sequence is given by the expression:\[a_n = 3n - 2\]
Key Concepts
Common Differencenth Term FormulaSequence Patterns
Common Difference
The common difference is an essential part of understanding arithmetic sequences. It is the constant amount that each term in the sequence increases or decreases from the previous term. In our current sequence, which is 1, 4, 7, 10, and so on, each number increases by 3. This 3 is the common difference.
To determine the common difference in a sequence, simply subtract any term from the term that follows it. For instance, in the sequence given:
To determine the common difference in a sequence, simply subtract any term from the term that follows it. For instance, in the sequence given:
- First term: 1, the second term: 4, so the difference: \(4 - 1 = 3\).
- Second term: 4, the third term: 7, so the difference: \(7 - 4 = 3\).
- Third term: 7, the fourth term: 10, so the difference: \(10 - 7 = 3\).
nth Term Formula
The nth term formula is a straightforward way to find any term of an arithmetic sequence without counting each term one by one. This formula is written as \(a_n = a_1 + (n-1) \times d\), where:\
- \(a_n\) is the nth term you are trying to find.
- \(a_1\) is the first term in the sequence.
- \(d\) is the common difference.
In the given sequence, we have already identified that \(a_1 = 1\) and the common difference \(d = 3\). Suppose you want to find the 5th term. Plug in the values to get:\[a_5 = 1 + (5-1) \times 3\] Simplifying, it becomes \[1 + 4 \times 3 = 13\]. Thus, the 5th term is 13.
This formula is invaluable for quickly finding terms further along in the sequence without manually calculating each one.
- \(a_n\) is the nth term you are trying to find.
- \(a_1\) is the first term in the sequence.
- \(d\) is the common difference.
In the given sequence, we have already identified that \(a_1 = 1\) and the common difference \(d = 3\). Suppose you want to find the 5th term. Plug in the values to get:\[a_5 = 1 + (5-1) \times 3\] Simplifying, it becomes \[1 + 4 \times 3 = 13\]. Thus, the 5th term is 13.
This formula is invaluable for quickly finding terms further along in the sequence without manually calculating each one.
Sequence Patterns
Recognizing sequence patterns is crucial for working with arithmetic sequences and beyond. In arithmetic sequences, patterns are defined by the constant common difference between terms. Observing these patterns allows us to generalize, predict future terms, and understand the sequence's behavior.
Identifying sequence patterns involves looking for regularities like our common difference. For example, you start with the first term and repeatedly add 3 to get the next terms. Observing sequence patterns:
Identifying sequence patterns involves looking for regularities like our common difference. For example, you start with the first term and repeatedly add 3 to get the next terms. Observing sequence patterns:
- Starting with 1, add 3 to get 4.
- Add 3 to 4 to get 7.
- Add 3 to 7 to continue to 10.
Other exercises in this chapter
Problem 25
Show that \(x-y\) is a factor of \(x^{n}-y^{n}\) for all natural numbers \(n .\) \(\left[\text { Hint: } x^{k+1}-y^{k+1}=x^{k}(x-y)+\left(x^{k}-y^{k}\right) y\r
View solution Problem 25
23-32 me the common difference, the fifth term, the \(n\)th term, and the 100th term of the arithmetic sequence. $$4,9,14,19, \dots$$
View solution Problem 26
Find the first four terms in the expansion of \(\left(x^{1 / 2}+1\right)^{30}\)
View solution Problem 26
Determine the common ratio, the fifth term, and the \(n\)th term of the geometric sequence. $$ 1, \sqrt{2}, 2,2 \sqrt{2}, \ldots $$
View solution