Problem 9 - 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: Understand the General Formula for the n-th Term

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

Step 2: Substitute Known Values into the Formula

For this sequence, \( a = 3 \) and \( r = 5 \). To find the fourth term, substitute these values and \( n=4 \) into the formula: \( a_4 = 3 imes 5^{(4-1)} \).

Step 3: Simplify the Expression for the Fourth Term

Calculate the exponent first: \( 5^{3} = 125 \). Then multiply by the first term: \( 3 imes 125 = 375 \).

Step 4: Conclusion

The fourth term of the geometric sequence is 375.

Key Concepts

nth Term Formula

In a geometric sequence, each term after the first is produced by multiplying the previous one by a fixed, non-zero number called the common ratio. To find any term in this sequence, we use the nth term formula: - The formula is given by \( a_n = a \times r^{(n-1)} \). - Here, \( a_n \) represents the term number \( n \), \( a \) is the first term, and \( r \) is the common ratio. This formula helps in determining any term in the sequence if you know the starting term and the common ratio, two critical variables in this mathematical pattern. It's a powerful tool because it allows you to dive straight into any part of the sequence without calculating each term one by one.

Common Ratio

The common ratio in a geometric sequence is crucial. It determines how each term in the sequence grows from the previous term. When you're given a sequence, you can find the common ratio by dividing any term by the term before it. For example: - If you have the sequence 3, 15, 75, ... - Divide 15 by 3 to get 5. - Divide 75 by 15 to get 5. Hence, the common ratio \( r \) is 5. When the common ratio is greater than 1, the sequence grows exponentially. In contrast, if \( 0 < r < 1 \), the sequence diminishes but remains above zero. A negative \( r \) can create alternating positive and negative terms.

First Term

The journey of any geometric sequence starts with the first term, noted as \( a \). It is the initial value from which all subsequent terms are built. Understanding the first term is essential because every calculation for finding any term in the sequence will start here. For instance, consider our example where \( a = 3 \). This signifies that 3 is the foundation of our sequence. With the common ratio of 5, each term is built by multiplying 3 with increasing powers of 5. The value of \( a \) sets an initial "scale" for the sequence, directly affecting all subsequent terms and their magnitudes.

Sequence Calculation

Sequence calculation involves substituting known values into the nth term formula to arrive at specific terms. Using our example, finding the fourth term involves a few straightforward steps. - With \( a = 3 \), \( r = 5 \) and \( n = 4 \), substitute these into the nth term formula: \( a_4 = 3 \times 5^{(4-1)} \). - Calculate \( 5^{3} \) which is 125. - Finally, multiply 3 and 125 to obtain 375. This method showcases how systematic substitution can yield precise results, enabling you to determine any term's value given the initial conditions of your sequence.