Problem 23 - Algebra 2

Question

Write an explicit formula for each sequence. Then find \(a_{12}\) $$ 2,5,10,17,26, \dots $$

Step-by-Step Solution

Answer

The explicit formula for the sequence is \(a_{n} = n^2 + 1\). The 12th term, \(a_{12}\), is 145.

Step 1: Identify the Sequence Pattern

To find the pattern, consider the difference between the successive terms. Each term appears to be 3 more than the product of the original position in the sequence (not zero-based) and the previous term. This pattern can be expressed as: \(a_{n} = n * a_{n-1} + 3\), where \(a_{n}\) is the nth term and \(a_{n-1}\) is the previous term.

Step 2: Write Down the Explicit Formula

The formula found in step 1 can be considered a recursive formula because it defines each term in terms of the previous one. To write this as an explicit formula (i.e., one that allows us to calculate terms directly without referencing earlier terms), note that each term equals the square of its position in the sequence (not zero-based) plus one. This can be expressed as: \(a_{n} = n^2 + 1\).

Step 3: Apply the Formula to Find \(a_{12}\)

Substitute 12 into the explicit formula: \(a_{12} = 12^2 + 1 = 144 + 1 = 145\).

Key Concepts

Recursive Formula

A recursive formula is a way of defining a sequence by expressing each term as a function of its preceding term. For our sequence, we noticed that

• The pattern involves a product and an addition operation.
• Each term depends on the one before it.

Specifically, the recursive formula given is:\[a_{n} = n \cdot a_{n-1} + 3\]This means each term is the result of multiplying its position in the sequence by the previous term and then adding 3. Recursive formulas are particularly useful because they mirror the sequence's inherent structure. They allow us to understand how the sequence grows from one term to the next.

Sequence Pattern

Understanding the sequence pattern is key to identifying how the sequence behaves. The pattern in a sequence is the consistent rule it follows as it progresses.

• For the sequence given: \(2, 5, 10, 17, 26, \ldots\), a noticeable feature is the increment in terms.
• To identify the pattern, consider how the terms change as they progress.

For this sequence, we see that the increase is not simply one fixed number but involves a mathematical operation related to the term's position or index. This recognition helps us convert the recursive pattern into an explicit formula.

Nth Term Calculation

The nth term calculation allows us to find any term in the sequence directly. This is quite powerful because it saves the hassle of calculating all preceding terms. The explicit formula removes the dependency on the previous term, allowing for efficient computation. In our case, the explicit formula is derived as:\[a_{n} = n^2 + 1\]

• This formula expresses each term as the square of its position plus one.

Using this rule, we substituted 12 into the formula to find\[a_{12} = 12^2 + 1 = 145\]This shows how simply knowing the explicit formula can provide quick answers, helping to find any term with minimal effort.