Problem 33 - Algebra and Trigonometry

Question

Find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 9th term is \(-5 ; 15\) th term is 31

Step-by-Step Solution

Answer

The first term is -53, and the common difference is 6. The recursive formula is \(a_n = a_{n-1} + 6\), and the nth term formula is \(a_n = 6n - 59\).

Step 1: Understand the Arithmetic Sequence

In an arithmetic sequence, each term increases by a constant value called the common difference. The nth term of an arithmetic sequence can be expressed as: \(a_n = a_1 + (n-1)d\) where \(a_n\) is the nth term, \(a_1\) is the first term, and \(d\) is the common difference.

Step 2: Set up Equations for the Given Terms

Given that the 9th term (\(a_9\)) is -5 and the 15th term (\(a_{15}\)) is 31, we can set up two equations using the formula for the nth term: \(a_9 = a_1 + 8d = -5\) \(a_{15} = a_1 + 14d = 31\).

Step 3: Solve the System of Equations

Subtract the first equation from the second to eliminate \(a_1\): \((a_1 + 14d) - (a_1 + 8d) = 31 - (-5)\) \(6d = 36\) \(d = 6\). Now substitute \(d = 6\) back into the first equation: \(a_1 + 8(6) = -5\) \(a_1 + 48 = -5\) \(a_1 = -5 - 48\) \(a_1 = -53\).

Step 4: Write the Recursive Formula

The recursive formula for an arithmetic sequence is given by: \(a_1 = -53\) \(a_{n} = a_{n-1} + d\) where \(d = 6\).

Step 5: Write the Formula for the nth Term

Using the formula for the nth term: \(a_n = a_1 + (n-1)d\) Substitute \(a_1 = -53\) and \(d = 6\): \(a_n = -53 + (n-1)6\) \(a_n = -53 + 6n - 6\) \(a_n = 6n - 59\).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference, denoted as \(d\), is the constant amount by which each term in the sequence increases or decreases from the previous term. It is critical to identifying and understanding the pattern of the sequence. To find the common difference, you can subtract any term from the term that follows it. In the given exercise, we had two terms: the 9th term \((a_9 = -5)\) and the 15th term \((a_{15} = 31)\).

• Form the two equations using the nth term formula.
• Solve to remove one variable and find the common difference.

First equation: \(a_9 = a_1 + 8d = -5\).
Second equation: \(a_{15} = a_1 + 14d = 31\).
Subtract the first equation from the second:
\begin{align*}(a_1 + 14d) - (a_1 + 8d) & = 31 - (-5)\6d & = 36\d & = 6.d = 6.This is the common difference.\end{align*}

Nth Term Formula

The nth term formula of an arithmetic sequence allows you to find the value of a specific term in the sequence without having to list out all the previous terms. The formula is given as:\[a_n = a_1 + (n-1)d\]Here,

• \(a\_n\): the nth term you want to find.
• \(a\_1\): the first term of the sequence.
• \(d\): the common difference between terms.
• \(n\): the position of the term in the sequence.

Using the given exercise, we had already calculated the common difference as \(d = 6\), and the first term as \(a\_1 = -53\). To find the nth term formula:Insert the values into the formula:\begin{align*}a\_n & = a\_1 + (n-1)d\a\_n & = -53 + (n-1) * 6\a\_n & = -53 + 6n - 6\a\_n & = 6n - 59.\end{align*}

Recursive Formula

The recursive formula provides a way to find each term in a sequence based on the previous term. It is particularly useful for sequences where the relationship between terms is a constant change, as is the case with arithmetic sequences. The general form of the recursive formula for an arithmetic sequence is:\[a_1 = \text{first term}\a_{n} = a_{n-1} + d \text{for } n > 1\]In the given problem, we have already found:

• The first term \(a\_1 = -53\).
• The common difference \(d = 6\).

So, the recursive formula for this sequence is:\[a_1 = -53\a_{n} = a_{n-1} + 6 \text{for } n > 1\]With this formula, you can find every term in the sequence by starting with \(a\_1\) and adding the common difference \(d\) to the previous term.