Problem 44 - Algebra 2

Question

a. Open-Ended Write four terms of a sequence of numbers that you can describe both recursively and explicitly. b. Write a recursive formula and an explicit formula for your sequence. c. Find the 20 th term of the sequence by evaluating one of your formulas. Use the other formula to check your work.

Step-by-Step Solution

Answer

The sequence defined is 1,3,5,7,...; the recursive and explicit formulas are \(a_{n} = a_{n-1} + 2\) (for \(n > 1\), \(a_1 = 1\)) and \(a_n = 2n - 1\), respectively; and the 20th term of the sequence is 39.

Step 1: Defining a sequence

Start by defining an easy sequence of numbers. Let's choose an arithmetic sequence where each term is 2 more than the previous term, starting with 1. Thus, the first four terms are 1, 3, 5, 7.

Step 2: Writing the recursive formula

The recursive formula for a sequence sets each term as a function of its preceding term or terms. For this sequence, since each term is 2 more than its predecessor, the recursive formula would be: \(a_{n} = a_{n-1} + 2\) for \(n > 1\), and \(a_1 = 1\) for the initial term.

Step 3: Writing the explicit formula

The explicit formula gives the nth term as a function of the number n. For the sequence we have chosen, since it's an arithmetic sequence, the explicit formula would be: \(a_n = 2n - 1\) for all positive integers n.

Step 4: Finding the 20th term of the sequence

Let's first use the explicit formula to work out the 20th term of the sequence. Using \(n = 20\) in \(a_n = 2n - 1\), we find \(a_{20} = 2*20 - 1 = 39\). This suggests that the 20th term of the sequence is 39.

Step 5: Checking the work

To verify, let's use the recursive formula to find the 20th term. This is more cumbersome because we would have to calculate all the previous terms up to the 19th term using the recursive formula. However, to simplify, we can just use the explicit formula to find the 19th term and then add 2 to it to get the 20th term. The 19th term according to the explicit formula is \(2*19 - 1 = 37\). Adding 2 gives us 39, which is the same as what we got from the explicit formula. This confirms that our work is correct.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. In simpler terms, each term after the first is obtained by adding a fixed number, called the common difference, to the previous term. For example, in the sequence 1, 3, 5, 7, the common difference is 2. Each term increases by 2 from the term before it.

This pattern is predictable, making it easy to calculate any term in the sequence if you know the first term and the common difference. Understanding this concept is key to forming both recursive and explicit formulas.

Recursive Formula

A recursive formula defines each term of a sequence relative to one or more preceding terms. It's like building blocks, where each block builds on the previous one.

For our arithmetic sequence starting from 1 and increasing by 2 each time, the recursive formula is expressed as:
\(a_{n} = a_{n-1} + 2\)
where \(n > 1\), and \(a_1 = 1\) for the initial term.

This formula tells us that to find the next term in the sequence, you simply take the previous term and add 2. Recursive formulas can be straightforward, but calculating terms far down the sequence one by one can become laborious.

Explicit Formula

An explicit formula allows you to calculate any term in the sequence directly, without needing to know the previous terms. This is particularly useful for finding terms further along in the sequence without having to count through each term sequentially.

For our chosen sequence, the explicit formula is:
\(a_n = 2n - 1\)
where \(n\) is the term number.

This formula makes it easy to find, say, the 20th term, by simply plugging in \(n = 20\) to get \(a_{20} = 39\). Explicit formulas are efficient for sequences with a known pattern, like arithmetic sequences.

Arithmetic Progression

Arithmetic progression is a term often used interchangeably with arithmetic sequence. It describes the same concept of a sequence where the difference between consecutive terms, called the common difference, remains constant.

An arithmetic progression can be visualized as a linear sequence of numbers:

• The first term, \(a_1\), is the starting point.
• Each subsequent term is obtained by adding the common difference to the previous term.

In our example, starting from 1 and adding 2 each time creates the sequence 1, 3, 5, 7, and so on.

Recognizing and understanding arithmetic progression is crucial for solving problems involving sequences, as it underpins both the creation and the use of recursive and explicit formulas.