Problem 52 - College Algebra with Corequisite Support

Question

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\left\\{-5,-\frac{10}{3},-\frac{5}{3}, \ldots\right\\} $$

Step-by-Step Solution

Answer

The explicit formula is \( a_n = \frac{5n-20}{3} \).

Step 1: Identify the First Term

The first term of the sequence is the first number given. In this sequence, the first term \( a_1 \) is \( -5 \).

Step 2: Determine the Common Difference

The common difference \( d \) in an arithmetic sequence is found by subtracting the first term from the second term. Here, the difference is \(-\frac{10}{3} - (-5)\). Change \(-5\) to a fraction with the same denominator: \(-5 = -\frac{15}{3}\). Now, \(d = -\frac{10}{3} - (-\frac{15}{3}) = \frac{5}{3}\).

Step 3: Write the Formula for the nth Term

The formula for the nth term \( a_n \) of an arithmetic sequence is given by \( a_n = a_1 + (n-1)d \). Substitute the values found: \( a_n = -5 + (n-1)\frac{5}{3} \).

Step 4: Simplify the Formula

Distribute \((n-1)\) within the equation: \( a_n = -5 + \frac{5(n-1)}{3} \). Simplify to get: \( a_n = -5 + \frac{5n}{3} - \frac{5}{3} \). Combine like terms: \( a_n = -\frac{15}{3} + \frac{5n}{3} - \frac{5}{3} = \frac{5n-20}{3} \).

Key Concepts

Explicit Formula

An explicit formula is a mathematical expression used to directly compute any term in an arithmetic sequence without needing to calculate all preceding terms. This is really useful because it saves time and effort, especially in sequences with a high number of terms.
In an arithmetic sequence, the explicit formula for the nth term is represented by:

• \( a_n = a_1 + (n-1)d \)

Where:

• \( a_n \) is the nth term.
• \( a_1 \) is the first term of the sequence.
• \( d \) is the common difference between consecutive terms.
• \( n \) is the term number you want to find.

The beauty of the explicit formula is that once the first term and common difference are identified, you can plug in any value for \( n \) to find the corresponding term. This formula is derived from recognizing that each term is consistently "adding on" the common difference based on its position in the sequence. As such, the explicit formula is widely appreciated for its straightforwardness.

Common Difference

The common difference is an essential aspect of an arithmetic sequence, signifying the constant amount that is added (or subtracted) to each term to get to the next. Finding this difference is crucial since it allows us to build the sequence step by step from the first term.
The common difference \( d \) is calculated by subtracting the first term from the second term.

• For the given sequence \( a = \{-5, -\frac{10}{3}, -\frac{5}{3}, \ldots\} \), determine \( d \) by evaluating: \( d = -\frac{10}{3} - (-5) \).

To make this calculation easier, it’s often helpful to rewrite terms to have the same denominator:

• \( -5 \) can be rewritten as \( -\frac{15}{3} \).

Now the calculation becomes straightforward:

• \( d = -\frac{10}{3} + \frac{15}{3} = \frac{5}{3} \)

Here, the positive \( \frac{5}{3} \) indicates that the sequence is increasing by \( \frac{5}{3} \) each step.

Nth Term Formula

The nth term formula in an arithmetic sequence is an expression used to find any particular term in the sequence. It's a helpful tool that lets you calculate the desired term directly using the sequence's starting point and the consistent interval of change between terms.
The generic form of the nth term formula is:

• \( a_n = a_1 + (n-1)d \)

For the given problem, let’s recap how we applied specific values:

• \( a_1 = -5 \)
• \( d = \frac{5}{3} \)

Putting these into the formula yields:

• \( a_n = -5 + (n-1)\frac{5}{3} \)

Continue to distribute and simplify:

• \( a_n = -5 + \frac{5n}{3} - \frac{5}{3} \)
• This simplifies further to: \( a_n = \frac{5n-20}{3} \)

This final formula, \( \frac{5n-20}{3} \), means you can calculate any even obscure term number in the sequence quickly without constructing the whole sequence up to that point.