Problem 5 - Algebra and Trigonometry

Question

Find the \(n\)th term of the geometric sequence with given first term \(a\) and common ratio \(r .\) What is the fourth term? $$ a=3, \quad r=5 $$

Step-by-Step Solution

Answer

The fourth term is 375.

Step 1: Understanding the Problem

We need to find the formula for the nth term of a geometric sequence and use it with the given values for the first term and common ratio to find a specific term: the fourth term.

Step 2: Formula for the nth Term

The general formula for the nth term of a geometric sequence is given by \( a_n = a imes r^{n-1} \), where \( a \) is the first term of the sequence, \( r \) is the common ratio, and \( n \) is the term number.

Step 3: Substitute Known Values

Substitute the given values into the formula, where \( a = 3 \) and \( r = 5 \). To find the fourth term, substitute \( n = 4 \) into the formula: \( a_4 = 3 \times 5^{4-1} \).

Step 4: Calculate the Power of r

Now calculate the power of \( r \), which is \( 5^{4-1} = 5^3 = 125 \).

Step 5: Calculate the Fourth Term

Multiply \( 125 \) by the first term. Therefore, \( a_4 = 3 \times 125 = 375 \).

Key Concepts

nth term

In a geometric sequence, every term is related to its previous term by multiplication with a constant known as the common ratio. The nth term is remarkably useful, especially when you need to directly find a particular element in the sequence without listing all preceding terms. To compute the nth term specifically, you use its position in the sequence, denoted as \(n\), to plug into the formula. This helps pinpoint which term you're looking at without unnecessary calculations.

• The "nth term" refers to the term position in a sequence.
• It provides a systematic way to calculate any term directly.
• The process saves time by avoiding sequential calculations.

Understanding the concept of the nth term gives you the power to navigate through sequences efficiently. You simply need to know the sequence's rule, typically governed by the common ratio, and which position, \(n\), you are seeking.

common ratio

The common ratio in a geometric sequence is the factor you multiply by to proceed from one term to the next. This ratio remains consistent throughout the sequence, which gives geometric sequences their distinctive properties. Understanding this concept is crucial because the entire sequence hinges on this ratio to maintain its pattern.

• The common ratio is constant for a specific geometric sequence.
• It determines the sequence’s rate of growth or decay.
• For positive \(r\), terms increase; for negative \(r\), terms alternate signs.

With our example where \(a = 3\) and \(r = 5\), the sequence progresses rapidly. Each term is five times its previous one, demonstrating swift exponential growth. If the common ratio was a fraction, however, the sequence would shrink.

formula for nth term

To find the nth term in a geometric sequence, you apply the formula \( a_n = a \times r^{n-1} \). This formula is key as it maps the first term and the sequence’s common ratio to any term in the sequence. It requires you to raise the common ratio \(r\) to the power of \(n-1\), reflecting the sequence's exponential nature.

• Start by identifying the first term \(a\) and the common ratio \(r\).
• Choose the desired position \(n\) for the term.
• Plug values into the formula: \(a_n = a \times r^{n-1}\).

For example, with \(a = 3\) and \(r = 5\), to find the fourth term \(n = 4\), compute \(a_4 = 3 \times 5^{3} = 375\). This formula allows calculation of any term efficiently, with just a few essential inputs and operations.