Problem 37 - College Algebra

Question

For the following exercises, write a recursive formula for each arithmetic sequence. $$ a=\left\\{\frac{1}{6},-\frac{11}{12},-2, \ldots\right\\} $$

Step-by-Step Solution

Answer

Recursive formula: \( a_1 = \frac{1}{6}, a_n = a_{n-1} - \frac{13}{12} \) for \( n \geq 2 \).

Step 1: Identify the First Term

The first term of the sequence is given as \( a_1 = \frac{1}{6} \).

Step 2: Calculate the Common Difference

The common difference (\( d \)) in an arithmetic sequence is found by subtracting the first term from the second term. \[ d = -\frac{11}{12} - \frac{1}{6} \]. To do this, convert \( \frac{1}{6} \) to a fraction with a denominator of 12: \( \frac{1}{6} = \frac{2}{12} \). Thus, \( d = -\frac{11}{12} - \frac{2}{12} = -\frac{13}{12} \).

Step 3: Write the Recursive Formula

A recursive formula for an arithmetic sequence is expressed as \( a_n = a_{n-1} + d \). Using the identified first term and common difference, the formula becomes \( a_n = a_{n-1} - \frac{13}{12} \).

Step 4: Complete the Recursive Formula with Initial Condition

The complete recursive formula must include the initial condition: \[ a_1 = \frac{1}{6} \] and \[ a_n = a_{n-1} - \frac{13}{12} \text{ for } n \geq 2 \].

Key Concepts

Recursive Formula

A recursive formula helps us define each term of a sequence based on the preceding term.
This is particularly useful in arithmetic sequences, where each term is generated by adding a fixed "common difference" to the previous term.
In general, the recursive formula for an arithmetic sequence can be written as:\[a_n = a_{n-1} + d\]Where:

• \(a_n\) is the \(n\)-th term of the sequence
• \(a_{n-1}\) is the \((n-1)\)-th term, or the previous term
• \(d\) is the common difference

This formula is key because it allows you to calculate any term in the sequence once the first term and the common difference are known.
In our example, the recursive formula was determined to be \(a_n = a_{n-1} - \frac{13}{12}\), illustrating that each subsequent term is \(-\frac{13}{12}\) less than its predecessor.

Common Difference

The "common difference" in an arithmetic sequence is the consistent amount by which terms increase or decrease as you move from one term to the next.
To find the common difference, subtract the first term from the second term, like we did in our exercise:\[d = -\frac{11}{12} - \frac{1}{6}\]Converting \(\frac{1}{6}\) to a compatible denominator with \(\frac{12}{12}\), we get \(\frac{2}{12}\).
Thus, the common difference becomes:\[d = -\frac{11}{12} - \frac{2}{12} = -\frac{13}{12}\]This value indicates how much you subtract from each term to reach the next in the sequence.
A common difference can be positive (increasing sequence) or negative, like in our example, which signifies a decreasing sequence.

Initial Condition

The "initial condition" in a sequence specifies the starting point of the sequence, typically denoted as the first term. It is crucial as it serves as the base value from which all other terms are calculated using the recursive formula.
In the provided exercise, the initial condition was given as:\[a_1 = \frac{1}{6}\]This statement simply means that the first term of our sequence is \(\frac{1}{6}\).
Without this initial condition, the recursive formula would not have a reference point to generate other terms of the sequence.
Hence, it is always paired with the recursive formula to form a complete mathematical description of the sequence.

First Term

The "first term" is the starting value of any sequence and is fundamental for setting the trajectory of the sequence.

• In arithmetic sequences, it is often designated as \(a_1\).
• This initial value works hand in hand with the common difference to generate the rest of the sequence.

To identify the first term, simply look at the sequence's initial value, which in this case is:\[a_1 = \frac{1}{6}\]Once you have the first term, along with the recursive formula, you can construct the entire sequence.
This provides a clear understanding of how sequences evolve, starting from the first term.