Problem 43 - Precalculus with Limits

Question

In Exercises 35 - 44, write an expression for the \( n \)th term of the geometric sequence. Then find the indicated term. \( a_1 = 500, r = 1.02, n = 40 \)

Step-by-Step Solution

Answer

Therefore, the 40th term of the geometric sequence is \( 500 \cdot (1.02)^{39} \).

Step 1: Write out the general formula

The \( n \)th term, \( a_n \), of a geometric sequence can be found with the formula \( a_n = a_1 \cdot r^{n-1} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

Step 2: Substitute the given values

By substituting the given values into the formula, you have \( a_n = 500 \cdot (1.02)^{40-1} \).

Step 3: Simplify the expression

Simplify the expression to get the value of the 40th term. So, \( a_{40} = 500 \cdot (1.02)^{39} \).

Key Concepts

Geometric Progression

A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. For instance, in the sequence 2, 4, 8, 16, each number is the result of multiplying the previous number by 2. This pattern of multiplication makes a geometric progression fundamentally different from an arithmetic progression, where you add a constant to get the next term. The ability to identify a geometric progression is important because it allows us to predict future terms and to calculate properties of the series, such as its sum.

When we speak of the 'n-th term' of a geometric sequence, we are referring to a specific element in the sequence that occupies the position 'n'. For example, the 3rd term in the sequence above is 8, and the 4th term is 16. A clear understanding of the geometric progression is crucial for working with growth or decay problems in fields as diverse as finance, physics, and biology, where this type of progression naturally appears.

Common Ratio

The common ratio of a geometric sequence is the factor by which consecutive terms multiply. If each term of the sequence is found by multiplying the previous term by the same number, that number is referred to as the common ratio, denoted by 'r'. For the sequence 3, 9, 27, 81, the common ratio is 3 because each term is three times the term before it.

Understanding the common ratio is vital for solving problems related to geometric sequences, as it is a key component in the formula used to find the terms of the sequence. To identify the common ratio, you can divide any term in the sequence by the preceding term. For example, if you have the sequence 5, 15, 45, dividing 15 by 5 or 45 by 15 will both yield the common ratio 3.

Sequence Term Calculation

To calculate the nth term of a geometric sequence, we use the formula: \( a_n = a_1 \times r^{(n-1)} \), where \( a_n \) is the nth term we wish to find, \( a_1 \) is the first term of the sequence, 'r' is the common ratio, and 'n' is the term's position in the sequence. By substituting the known values into this formula, we can determine any term's value.

For instance, with a first term \( a_1 \) of 500 and a common ratio 'r' of 1.02, to find the 40th term, we denote 'n' as 40 and use the formula to compute \( a_{40} = 500 \times (1.02)^{39} \). Through this calculation, the monumental power of geometric sequences becomes apparent, as we can see how a small change in the common ratio compounds dramatically over a large number of terms, which is especially relevant in the contexts of compound interest and exponential growth.