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 formula for the nth term is \(a_n = 3n - 2\).
1Step 1: Observe the Pattern
First, identify any pattern by looking at the given sequence: 1, 4, 7, 10, .... It seems there is a regular increase from one term to the next.
2Step 2: Calculate the Common Difference
Subtract each term from the following term to find the common difference: \(4 - 1 = 3\), \(7 - 4 = 3\), \(10 - 7 = 3\). The common difference is 3.
3Step 3: Use the Formula for the Arithmetic Sequence
An arithmetic sequence is given by the formula \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term and \(d\) is the common difference.
4Step 4: Substitute the Known Values into the Formula
Substitute \(a_1 = 1\) (first term of the sequence) and \(d = 3\) (the common difference) into the formula: \(a_n = 1 + (n-1) \times 3\).
5Step 5: Simplify the Formula
Simplify the expression by distributing the multiplication: \(a_n = 1 + 3n - 3\), which simplifies to \(a_n = 3n - 2\).
Key Concepts
Common DifferenceNth Term FormulaSequence Pattern Recognition
Common Difference
In an arithmetic sequence, the common difference is the consistent amount that each term increases by. Recognizing the common difference is the key to understanding how the sequence grows and predicting future terms.
In our example, the sequence is 1, 4, 7, 10,... To find the common difference, we subtract a term from the next term:
In our example, the sequence is 1, 4, 7, 10,... To find the common difference, we subtract a term from the next term:
- Subtract 1 from 4, which gives you 3.
- Subtract 4 from 7, also 3.
- Subtract 7 from 10, 3 again.
Nth Term Formula
The nth term formula is a powerful tool for finding any term in an arithmetic sequence without listing all previous terms. Once you have identified the common difference, you can use the formula:
\[ a_n = a_1 + (n-1)d \]
- Here, \(a_1\) represents the first term, and \(d\) is the common difference.
- In our sequence, \(a_1 = 1\) and \(d = 3\). Substituting these values into the formula, we have:
\[ a_n = 1 + (n-1) imes 3 \]
Simplifying further, by distributing 3 and combining like terms, we get:
\[ a_n = 1 + 3n - 3 \]
\[ a_n = 3n - 2 \]
This formula enables you to find the nth term directly by substituting the value of \(n\). For instance, if you wish to find the 5th term, just substitute \(n = 5\) into the formula.
\[ a_n = a_1 + (n-1)d \]
- Here, \(a_1\) represents the first term, and \(d\) is the common difference.
- In our sequence, \(a_1 = 1\) and \(d = 3\). Substituting these values into the formula, we have:
\[ a_n = 1 + (n-1) imes 3 \]
Simplifying further, by distributing 3 and combining like terms, we get:
\[ a_n = 1 + 3n - 3 \]
\[ a_n = 3n - 2 \]
This formula enables you to find the nth term directly by substituting the value of \(n\). For instance, if you wish to find the 5th term, just substitute \(n = 5\) into the formula.
Sequence Pattern Recognition
Recognizing patterns in sequences is crucial in mathematics, as it allows us to predict how sequences will continue. In arithmetic sequences, recognizing patterns helps identify the regular interval increase between terms.
In the original sequence 1, 4, 7, 10,..., it is evident that the terms increase by the same amount each time. The pattern here is straightforward linear growth by 3.
This practice of pattern recognition not only aids in calculating the common difference but also in understanding the underlying structure of the sequence. You can apply these patterns universally across arithmetic sequences, which helps when dealing with more complex sequences or missing terms. Always start by identifying how much each term changes relative to the next and use it as the foundation of further calculations. Getting familiar with these patterns develops your mathematical intuition, making arithmetic progression problems less daunting.
In the original sequence 1, 4, 7, 10,..., it is evident that the terms increase by the same amount each time. The pattern here is straightforward linear growth by 3.
This practice of pattern recognition not only aids in calculating the common difference but also in understanding the underlying structure of the sequence. You can apply these patterns universally across arithmetic sequences, which helps when dealing with more complex sequences or missing terms. Always start by identifying how much each term changes relative to the next and use it as the foundation of further calculations. Getting familiar with these patterns develops your mathematical intuition, making arithmetic progression problems less daunting.
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\righ
View solution Problem 25
Determine the common ratio, the fifth term, and the \(n\) th term of the geometric sequence. $$0.3,-0.09,0.027,-0.0081, \ldots$$
View solution Problem 26
Determine the common difference, the fifth term, the \(n\) th term, and the 100 th term of the arithmetic sequence. $$11,8,5,2, \dots$$
View solution Problem 26
Find the first four terms in the expansion of \(\left(x^{1 / 2}+1\right)^{30}\)
View solution