Problem 44 - 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. $$ 3,6,11,18,27, \dots $$

Step-by-Step Solution

Answer

The next two terms in the sequence are 38 and 51 respectively. The explicit formula for the \(n\) th term is \(n^2+2\), and the recursive formula is \(a_n = a_{n-1} + 2n+1\). So, the formula can be classified as both explicit and recursive.

Step 1: Identify the pattern

Calculate the difference between consecutive terms. The differences are \(6-3=3\), \(11-6=5\), \(18-11=7\), \(27-18=9\). The differences between the terms are increasing by 2.

Step 2: Calculate next two terms

Since the differences are increasing by 2 each time, the difference for the next term will be \(9+2=11\). Adding this to the previous term \(27+11\) gives the next term, which is 38. The same step is repeated to get the next term 50 (difference is \(11+2=13\), \(38+13=51\). Thus, the next two terms in the sequence are 38 and 51.

Step 3: Formulate the \(n\) th term

By observing the pattern (difference increases by 2 each term), we can derive an explicit formula from the sequence. Since the first term is 3 and the increasing difference (3, 5, 7, 9) are of the form \(2n+1\), the explicit formula for the sequence is \(n^2+2\). In the recursive form, the formula becomes \(a_n = a_{n-1} + 2n+1\), where \(a_n\) is the \(n\) th term and \(a_{n-1}\) is the previous term.

Step 4: Identify the formula type

The formula derived in step 3 could be written in two different forms. The explicit form is \(n^2+2\), while the recursive form is \(a_n = a_{n-1} + 2n+1\). So, the formula is both explicit and recursive.

Key Concepts

Understanding Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference. For example, in the sequence 3, 6, 9, 12, the common difference is 3. Arithmetic sequences are prevalent in mathematics due to their straightforward structure and predictability.

In our provided sequence (3, 6, 11, 18, 27, ...), the difference changes each time, so this specific sequence is not an arithmetic sequence. However, understanding arithmetic sequences is crucial as it forms the base for many other types of sequences, including this one, where differences have a consistent pattern or growth.

• An arithmetic sequence will have a formula of the form: \(a_n = a_1 + (n-1)d\)
• Where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term position.

Exploring the Recursive Formula

A recursive formula is a formula that defines each term of a sequence using the preceding term or terms. This form of formula is incredibly useful for understanding the relationship between successive terms in a sequence.

For our sequence (3, 6, 11, 18, 27, ...), the recursive formula is determined by recognizing how each term is derived from the previous one. Once a pattern is established—in this case, where each new term's increment increases by 2 ( 3, 5, 7, 9)—we can create a recursive formula.

• The recursive formula derived from the sequence: \(a_n = a_{n-1} + 2n + 1\)
• In this formula, \(a_{n-1}\) represents the previous term, and \(2n + 1\) represents the increasing difference.

This type of formula is typically more intuitive but can be cumbersome if we want to calculate terms that are far along in the sequence.

Deriving the Explicit Formula

The explicit formula provides a direct way to calculate any term in the sequence without referencing previous terms. This is advantageous in sequences with complex changes or when needing to rapidly calculate terms far into the sequence.

For our sequence, recognizing the change pattern allowed for an explicit formula: \(a_n = n^2 + 2\). This derives from an observation that the differences between terms form a sequence themselves, ultimately leading to a pattern based on squares plus a constant.

• The benefits of explicit formulas include efficiency and transparency.
• This allows for quick computations and easier recognition of underlying sequence patterns.

In our case, you can calculate any term by simply substituting \(n\) in the formula.