Problem 34 - Algebra and Trigonometry

Question

Find the nth term \(a_{n}\) of each geometric sequence. When given, \(r\) is the common ratio. $$ 5,10,20,40, \ldots $$

Step-by-Step Solution

Answer

The nth term \( a_n \) is given by \ 5 \times 2^{(n-1)} \.

Step 1: Identify the first term (a₁)

The first term of the geometric sequence is the first number in the given series. Here, the first term, \(a_1\), is 5.

Step 2: Find the common ratio (r)

The common ratio, \(r\), is found by dividing the second term by the first term. Calculate \(r\) using \( \frac{a_2}{a_1} \): \[\frac{10}{5} = 2\]. Therefore, \( r = 2 \).

Step 3: Use the formula for the nth term of a geometric sequence

The formula for the nth term of a geometric sequence is given by \( a_n = a_1 \times r^{(n-1)} \).

Step 4: Plug in the values

Substitute \( a_1 = 5, \ r = 2 \) into the formula: \ a_n = 5 \times 2^{(n-1)} \.

Key Concepts

First Term

The first term of a geometric sequence is the initial number from which the sequence begins. In a geometric sequence, each term is generated by multiplying the previous term by a constant number called the common ratio. Identifying the first term is crucial as it sets the stage for the entire sequence. In the given sequence, 5, 10, 20, 40, the first term, denoted as \(a_1\), is 5. This term is represented as the starting point of the sequence and is essential in calculating any subsequent terms.

Common Ratio

The common ratio is a key element that determines how the terms in a geometric sequence progress. It is the constant factor between any two consecutive terms. To find the common ratio, divide the second term by the first term. For example, in the sequence 5, 10, 20, 40, the common ratio \(r\) is calculated as follows:

• Second term \(a_2\) = 10
• First term \(a_1\) = 5

So, \( r = \frac{10}{5} = 2 \).
This means each term is multiplied by 2 to get to the next term in the sequence. Understanding the common ratio makes it easy to predict future terms.

Nth Term Formula

The nth term formula is used to determine any specific term in a geometric sequence without listing all previous terms. This formula incorporates the first term and the common ratio to calculate the value of the term at any position \(n\). The formula is given by:
\[ a_n = a_1 \times r^{(n-1)} \]
Where:

• \(a_n\) is the nth term
• \(a_1\) is the first term
• \(r\) is the common ratio
• \(n\) is the term number in the sequence

For example, to find the 4th term in the sequence 5, 10, 20, 40, with \(a_1 = 5\) and \(r = 2\), plug in the values into the formula:
\[ a_4 = 5 \times 2^{(4-1)} = 5 \times 2^3 = 5 \times 8 = 40 \]
This makes it easy to see how the formula works and helps in quickly finding any term in the sequence. Always substitute the known values correctly to avoid errors in computation.