Problem 42 - Algebra 2

Question

Find the next two terms in each sequence. Write a formula for the \(n\) th term. Identify each formula as explicit or recursive. $$ 1,3,5,7,9, \dots $$

Step-by-Step Solution

Answer

The next two terms of the sequence are 11 and 13. The explicit formula for the nth term of the sequence is An = 2n - 1.

Step 1: Identify The Pattern

Analyze the sequence and identify the pattern. In the sequence 1,3,5,7,9 ... it's clear that each term is increased by 2 from the previous term. This is an arithmetic progression where the common difference (d) is 2.

Step 2: Find The Next Two Terms

To find the next two terms, add the common difference (2) to the last two given terms respectively. Thus, the next term after 9 is 9+2=11 and the term following that is 11+2=13. So, the next two terms of the sequence are 11 and 13.

Step 3: Find A Formula For The nth Term

The nth term (An) of an arithmetic sequence can be found by the formula: An = A1 + (n-1)d, where A1 is the first term and d is the common difference. Substituting the given values into the formula gives: An = 1 + (n-1)2 = 2n-1.

Step 4: Identify The Formula

The obtained formula for the nth term, An = 2n - 1, is an explicit formula as it expresses the nth term in terms of an arithmetic expression involving n, and it directly computes any term of the sequence without the need to find the previous terms.

Key Concepts

nth term formula

In arithmetic sequences, finding the nth term is essential to understanding the sequence's behavior. The nth term formula provides a straightforward way to calculate any term in the sequence without listing all prior terms. For our sequence, the formula is given by the expression:

• \( A_n = A_1 + (n-1)d \)

Here, \( A_1 \) is the first term of the sequence, \( d \) is the common difference between consecutive terms, and \( n \) is the term number.
In our sequence (1, 3, 5, 7, 9), the first term \( A_1 \) is 1, and the common difference \( d \) is 2. Substituting these values, the nth term formula becomes:

• \( A_n = 1 + (n-1) imes 2 \)
• \( A_n = 2n - 1 \)

This formula helps in calculating the value of any term in the sequence without having to find all the preceding terms.

explicit formula

An explicit formula in arithmetic sequences allows for direct computation of any term in the sequence using the term number. Unlike recursive formulas, explicit formulas do not rely on previous terms to find the next term.
The explicit formula takes the form:

• \( A_n = A_1 + (n-1)d \)

This formula makes it easy to compute the nth term directly; thus, it is often preferred for ease of use and speed.
For the sequence we are discussing (1, 3, 5, 7, 9), the explicit formula we derived is:

• \( A_n = 2n - 1 \)

This compact form provides a robust method to find the value of any term without additional steps or computations. Unlike recursive formulas, explicit formulas are simpler and more efficient for finding terms at arbitrary positions in the sequence.

recursive formula

In contrast to the explicit formula, a recursive formula defines each term based on the previous term. This type of formula is useful when you want to understand the progression between terms rather than jumping straight to an arbitrary term. The recursive formula for arithmetic sequences can be defined as:

• \( A_n = A_{n-1} + d \)

Where \( A_{n-1} \) is the previous term and \( d \) is the common difference.
In the sequence (1, 3, 5, 7, 9), the recursive formula becomes:

• \( A_n = A_{n-1} + 2 \)

With \( A_1 = 1 \) as the starting point. To find any term using the recursive formula, knowledge of the preceding term is essential. This means you must "build up" the sequence term by term to reach your desired \( n^{th} \) term.
Despite its limitations in direct term finding, the recursive approach does provide insight into the step-by-step changes within a sequence, which can be valuable for certain analyses and applications.