Question

In Problems 22-25, find the first four terms of the sequence and a formula for the general term. $$ a_{n}=a_{n-1}+2 ; a_{1}=1 $$

Short answer

Question: Find the first four terms and the general term of the sequence defined by the recursive relation \(a_n = a_{n-1} + 2\), where \(a_1 = 1\). Answer: The first four terms are \(1, 3, 5, 7\), and the general formula for the sequence is \(a_n = 2n - 1\).

Step-by-step solution

Find the first term#a_1

The first term \(a_1\) is given as \(1\).

Find the second term#a_2

Use the recursive formula \(a_n = a_{n-1} + 2\) by plugging in \(n = 2\) to find \(a_2\). $$ a_2 = a_{2-1} + 2 = a_1 + 2 = 1 + 2 = 3 $$ So, the second term \(a_2\) is \(3\).

Find the third term#a_3

Now, find \(a_3\) by plugging \(n = 3\) into the recursive formula: $$ a_3 = a_{3-1} + 2 = a_{2} + 2 = 3 + 2 = 5 $$ The third term \(a_3\) is \(5\).

Find the fourth term#a_4

Again, use the recursive formula to find \(a_4\) by plugging in \(n = 4\): $$ a_4 = a_{4-1} + 2 = a_{3} + 2 = 5 + 2 = 7 $$ The fourth term \(a_4\) is \(7\).

Determine the general formula for \(a_n\)

To determine the general formula, let's examine the differences between consecutive terms: $$ a_2 - a_1 = 3 - 1 = 2 \\ a_3 - a_2 = 5 - 3 = 2 \\ a_4 - a_3 = 7 - 5 = 2 \\ $$ The difference between consecutive terms is constant, which indicates that the sequence is an arithmetic sequence. The common difference is \(2\), and the first term is \(a_1=1\). The general formula for an arithmetic sequence is: $$ a_n = a_1 + (n - 1)d $$ where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number. In this problem, \(a_1 = 1\) and \(d = 2\). So, the general formula for the sequence is: $$ a_n = 1 + (n - 1)2 $$ This simplifies to: $$ a_n = 2n - 1 $$ In conclusion, the first four terms of the sequence are \(1, 3, 5, 7\), and the general formula for the sequence is \(a_n = 2n - 1\).

Key concepts

Recursive Formula

A recursive formula is a method to define the terms of a sequence using the previous terms. It's like a set of instructions that you follow to find the next number in the sequence. For an arithmetic sequence, the recursive formula tells us how to get from one term to the next by adding a constant.

In the given problem, the recursive formula is \( a_n = a_{n-1} + 2 \). This indicates that each term in the sequence is obtained by taking the previous term and adding 2 to it. For instance, if you start with the first term \( a_1 = 1 \), you can use the formula to find the next terms:

• Second term: \( a_2 = a_1 + 2 = 3 \)
• Third term: \( a_3 = a_2 + 2 = 5 \)
• Fourth term: \( a_4 = a_3 + 2 = 7 \)

Recursive formulas are particularly useful when you need to find successive terms without needing to calculate a general term directly.

General Term

The general term of a sequence provides a formula to find any term in that sequence without needing to calculate all the preceding terms. For arithmetic sequences, which have a constant difference between consecutive terms, the general term is derived from a specific position in the sequence.

The general formula for an arithmetic sequence is \( a_n = a_1 + (n - 1)d \). Here, \( a_1 \) is the first term, \( n \) is the term position, and \( d \) is the common difference. In our problem, since \( a_1 = 1 \) and the common difference \( d \) is 2, we can express the general term as:\[ a_n = 1 + (n - 1) \times 2 \]This simplifies to \( a_n = 2n - 1 \). This formula is a powerful tool because you can find any term in the sequence, like the 50th term, just by plugging \( n = 50 \) into the formula, without needing to list all the previous terms.

Common Difference

The common difference is an essential part of an arithmetic sequence. It is the consistent difference between each pair of consecutive terms. Think of the common difference as the amount you "add" to get from one term to the next.

In the given sequence, we observed in the step-by-step solution that each term increases by 2 from the previous term:

• \( a_2 - a_1 = 3 - 1 = 2 \)
• \( a_3 - a_2 = 5 - 3 = 2 \)
• \( a_4 - a_3 = 7 - 5 = 2 \)

The consistent result here, 2, is our common difference, \( d \). This uniform addition is what defines the regular pattern of an arithmetic sequence, making it predictable and easy to work with. By knowing this difference, along with the first term, anyone can determine the entire sequence. It serves as the backbone of both the recursive and general formulas.