Problem 9 - Precalculus: Mathematics for Calculus

Question

Find the \(n\) th term of the geometric sequence with given first term \(a\) and common ratio \(r .\) What is the fourth term? $$a=7, \quad r=4$$

Step-by-Step Solution

Answer

The fourth term of the sequence is 448.

Step 1: Understanding the Formula for the n-th Term

The formula to find the n-th term of a geometric sequence is \( a_n = a \cdot r^{(n-1)} \), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number we are looking for.

Step 2: Substituting Known Values

We need to find the fourth term \(a_4\). Using the formula \( a_n = a \cdot r^{(n-1)} \), substitute \(a = 7\), \(r = 4\), and \(n = 4\). The expression becomes \( a_4 = 7 \cdot 4^{(4-1)} \).

Step 3: Calculating the Power

Calculate \( 4^{3} \). Since \( 4 \times 4 \times 4 = 64 \), we have \(4^3 = 64\).

Step 4: Finding the Fourth Term

Multiply the result of \(4^3\) by the first term \(a\): calculate \( 7 \times 64 \). This gives you \( 448 \).

Step 5: Conclusion

Thus, the fourth term \(a_4\) of the geometric sequence is 448.

Key Concepts

nth term of a sequence

In any geometric sequence, understanding how to find the nth term is crucial. A sequence is essentially a list of numbers, and each number in the list is called a term. The nth term refers to any specific term in that sequence. The ability to find this term allows you to identify any given position of a geometric sequence without needing to list out all the preceding numbers.

The formula for finding the nth term, denoted as \( a_n \), is \( a_n = a \cdot r^{(n-1)} \). Here, \( a \) represents the first term of the sequence, \( r \) is the common ratio, and \( n \) is the position of the term you want to find. For instance, if you want the 4th term, replace \( n \) with 4 in the formula. This gives a quick and efficient way to see how far a sequence has progressed at any specific point.

common ratio

The common ratio in a geometric sequence is a key element that connects all the terms. It's the constant factor that each term is multiplied by to get the next term. This ratio is the reason behind the consistent and predictable change between terms in a geometric sequence.

To find the common ratio, you can divide any term in the sequence by the previous one. For instance, with a first term \( a = 7 \) and common ratio \( r = 4 \), each subsequent term is four times the previous term. Common ratios can be whole numbers, fractions, or even negative numbers; whatever keeps the consistent multiplication pattern. Understanding this property helps in quickly identifying the audio sequence and predicting future terms efficiently.

geometric progression formula

The geometric progression formula is an essential tool in sequence mathematics. This formula, \( a_n = a \cdot r^{(n-1)} \), provides a systematic way to find any term of a geometric sequence.

Geometric progressions are patterns of growth often found in various real-life situations, such as calculating interest, population growth, or even in geometric shapes and fractals. By using the formula, the calculation becomes straightforward. You only need to know the first term \( a \) and the common ratio \( r \). Then, raise \( r \) to the power of \((n-1)\) to account for its multiplicative effect through the sequence and multiply it by \( a \) to retrieve the nth term. This formula distills the entire process of finding terms in a sequence down to a simple mathematical operation.