Problem 29 - Algebra and Trigonometry

Question

Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$ a_{1}=-20, d=-4 $$

Step-by-Step Solution

Answer

The 20th term of the sequence (\(a_{20}\)) is -96.

Step 1: Arithmetic Sequence Formula

In an arithmetic sequence the difference between any two consecutive terms is constant. This difference is referred to as the 'common difference' (\(d\)). The first term is denoted by \(a_{1}\). The general formula for the nth term of an arithmetic sequence can be stated as: \(a_{n} = a_{1} + (n - 1) * d\). Here, \(a_{1} = -20\) and \(d = -4\).

Step 2: Substitute Into the Formula

Now that we have the formula and the given values, we can substitute them into the formula to find \(a_{20}\). So, replacing \(a_{1}\) with -20, \(d\) with -4, and \(n\) with 20 in the formula: \(a_{20} = -20 + (20 - 1)*-4\).

Step 3: Solve the Equation

First, simplify the expression within the parentheses: \(a_{20} = -20 + (19)*-4\). Then, multiply: \(a_{20} = -20 + (-76)\). Adding -20 and -76 gives: \(a_{20} = -96\).

Key Concepts

Understanding the 'Common Difference'

In an arithmetic sequence, each term increases or decreases by the same amount from one term to the next. This consistent change is what we refer to as the 'common difference' and is denoted by the letter \(d\).

For instance, if the first term of a sequence is \(-20\) and the common difference is \(-4\), you would subtract 4 from each term to get the next one. This means:

• The second term would be \(-20 - 4 = -24\)
• The third term \(-24 - 4 = -28\)
• And so on.

The magic of the common difference lies in its simplicity. It forms the backbone of understanding arithmetic sequences and helps to forecast any term without calculating them one by one.

Exploring the 'General Term Formula'

The general term formula provides a shortcut to find any term in an arithmetic sequence. Rather than calculating each term sequentially, this formula empowers you to leap directly to any term, leapfrogging the earlier ones.

The formula is: \(a_{n} = a_{1} + (n - 1) \cdot d\).

In this equation:

• \(a_{n}\) represents the term you're interested in
• \(a_{1}\) is the first term
• \(d\) is the common difference, and
• \(n\) is the position of the term in the sequence.

Applying this formula can save time and effort, especially in large sequences. For example, with a first term of \(-20\) and a common difference of \(-4\), this formula can easily calculate any nth term, without requiring you to write out each preceding term.

Performing the 'nth Term Calculation'

Once you understand the components of the general term formula, performing the nth term calculation becomes straightforward. Let's explore it with our sequence where \(a_{1} = -20\) and \(d = -4\). We need to find the 20th term.

Using the formula: \[ a_{20} = a_{1} + (n - 1) \cdot d \] Substitute the known values: \(-20 + (20 - 1) \cdot (-4)\).

First, calculate the expression inside the parentheses: \(20 - 1 = 19\).

Next, multiply: \(19 \times (-4) = -76\).

Finally, add the first term: \(-20 + (-76) = -96\).

Therefore, the 20th term is \(-96\). This calculation illustrates the efficiency of the technique, eliminating the need to manually compute every term up to the one we are interested in.