Problem 28 - Precalculus Mathematics for Calculus

Question

Find the nth term of a sequence whose first several terms are given. $$5,-25,125,-625, \dots$$

Step-by-Step Solution

Answer

The nth term is \( a_n = 5 (-5)^{(n-1)} \).

Step 1: Identify the Pattern

Look at the given sequence: 5, -25, 125, -625. Observe that each term is a result of multiplication from the previous term.

Step 2: Determine the Common Ratio

Calculate the ratio between consecutive terms to find the common ratio (r). Divide the second term by the first term: \(-25/5 = -5\). Verify this ratio by dividing the third term by the second: \(125/-25 = -5\) and the fourth term by the third \(-625/125 = -5\). Thus, the common ratio is \(-5\).

Step 3: Use the Formula for the nth term

The nth term of a geometric sequence can be found using the formula \( a_n = a_1 \, r^{(n-1)} \) where \(a_1\) is the first term and \(r\) is the common ratio. In this sequence, \(a_1 = 5\) and \(r = -5\).

Step 4: Derive the nth Term Formula

Substitute the known values into the formula: \( a_n = 5 \, (-5)^{(n-1)} \). This gives the nth term formula for the sequence.

Key Concepts

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This sequence follows a specific pattern of multiplication. It is important because it helps us recognize patterns involving exponential growth or decay in various fields such as finance, physics, and population studies.

• Each term in the sequence is the product of its preceding term and the common ratio.
• The sequence can be both increasing or decreasing.
• A geometric sequence can have either positive or negative terms depending primarily on its common ratio.

In our example sequence, the terms are 5, -25, 125, -625, and so on. We observe that to go from one term to the next, we multiply by the common ratio, which we'll explore further in the next section.

Common Ratio

The common ratio is a key element in understanding geometric sequences. It is the constant factor you multiply by to get from one term to the next. Determining the common ratio helps us confirm the type of sequence we're working with and enables us to predict future terms.

• To find the common ratio, divide any term by its preceding term.
• Ensure consistency by checking multiple consecutive terms.
• A positive common ratio results in all terms retaining the same sign as the first term, while a negative ratio causes alternating signs.

For the sequence provided, the common ratio is calculated by dividing -25 by 5, resulting in a common ratio of -5. This indicates that each term is five times larger in absolute value but opposite in sign compared to the previous term, creating an alternating pattern of positive and negative numbers.

Sequence Formula

The sequence formula is a vital tool in predicting any term within a geometric sequence, known as the nth term. For a geometric sequence, the nth term can be calculated using the formula: \[ a_n = a_1 \times r^{(n-1)} \]Where:

• \(a_n\) is the nth term you're trying to find.
• \(a_1\) is the first term in the sequence.
• \(r\) is the common ratio.
• \((n-1)\) is the exponent indicating the position of the term in the sequence.

In our example, the first term \(a_1\) is 5 and the common ratio \(r\) is -5. By plugging these values into the formula, as you compute \[a_n = 5 \times (-5)^{(n-1)}\], you can find any term in the sequence by substituting the specific \(n\). This formula efficiently allows determining any number in the sequence without computing each of the preceding terms.