Problem 38 - Algebra and Trigonometry

Question

Find the nth term \(a_{n}\) of each geometric sequence. When given, \(r\) is the common ratio. $$ a_{2}=7 ; \quad r=\frac{1}{3} $$

Step-by-Step Solution

Answer

\[a_{n} = 21 \times \left( \frac{1}{3} \right)^{(n-1)}\]

Step 1: Identify the general formula

The nth term of a geometric sequence can be found using the formula: \[a_{n} = a_{1} \times r^{(n-1)}\]where \(a_{1}\) is the first term and \(r\) is the common ratio.

Step 2: Use given information for the second term

We know that \(a_{2} = 7\) and \(r = \frac{1}{3}\). Plug these into the formula for the second term: \[a_{2} = a_{1} \times r^{(2-1)}\]This simplifies to: \[7 = a_{1} \times \frac{1}{3}\].

Step 3: Solve for the first term \(a_{1}\)

To find \(a_{1}\), solve the equation: \[7 = a_{1} \times \frac{1}{3}\]Multiply both sides by 3 to isolate \(a_{1}\): \[a_{1} = 21\].

Step 4: Substitute back to find the nth term

Now that we have \(a_{1} = 21\), use the general formula again to find \(a_{n}\): \[a_{n} = 21 \times \left(\frac{1}{3}\right)^{(n-1)}\].

Key Concepts

nth term formula

In a geometric sequence, the nth term is found using a specific formula. This formula helps us determine any term in the sequence based on its position. The formula is: \(a_{n} = a_{1} \times r^{(n-1)}\). Here, \(a_{n}\) is the nth term, \(a_{1}\) is the first term, and \(r\) is the common ratio. The common ratio is raised to the power of \((n-1)\), which means you're multiplying the first term by the common ratio repeatedly for \(n-1\) times. This formula is crucial because it provides a quick way to find any term in the sequence without listing all previous terms.

common ratio

The common ratio, often represented as \(r\), is a key component of geometric sequences. It's the factor by which each term is multiplied to get the next term. For example, if \(a_{2} = 7\) and \(r = \frac{1}{3}\), it means that each term is one-third of the previous term. Understanding the common ratio helps us see the pattern in a geometric sequence and apply the nth term formula correctly. To find the common ratio in a sequence, you can divide any term by the previous term: \(r = \frac{a_{n}}{a_{n-1}}\).

first term calculation

Determining the first term \(a_{1}\) of a geometric sequence is often a necessary step in solving problems. For instance, from the given exercise, you start with the information \(a_{2} = 7\) and \(r = \frac{1}{3}\). Using the nth term formula, set up the equation for the second term: \(a_{2} = a_{1} \times \frac{1}{3}\). Simplify to find \(a_{1}\): \(7 = a_{1} \times \frac{1}{3}\). Multiply both sides by 3: \(a_{1} = 21\). Now you know the first term, which is essential for using the nth term formula to find any other term in the sequence.