Problem 68 - Algebra 2

Question

The sixth term in a geometric sequence is \(120 .\) The seventh term is \(40 .\) What is the first term in the sequence?

Step-by-Step Solution

Answer

The first term in the sequence is 29160.

Step 1: Identifying known and unknown variables

In this problem, you have the 6th term \(T6=120\) and the 7th term \(T7=40\). Your goal is to find the first term \(T1\) of the sequence.

Step 2: Determining the common ratio

The ratio between two consecutive terms in a geometric sequence is consistent. Hence, the common ratio \(r\) can be calculated as \(r=T7/T6=40/120=1/3\).

Step 3: Calculating the first term

Since the formula for the \(n-th\) term of a geometric sequence is \(ar^{(n-1)}\), we can use the calculated ratio and the given 6th term to solve for \(a\). This gives \(a=T6/r^{(5)}=120/(1/3)^{5}=120*243=29160\).

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio is the constant factor that links all terms in the sequence. To find the common ratio, you divide a term by its preceding term. This ratio remains the same for the entire sequence, providing a scalable relationship between terms. For instance, in the exercise we are solving, the 7th term is 40 and the 6th term is 120. By dividing these, we calculate the common ratio as follows:

• Common Ratio, \( r = \frac{T7}{T6} = \frac{40}{120} = \frac{1}{3} \)

This means each term is one-third the value of the previous term. The concept of a common ratio is crucial because it maintains the sequence's structure by proportionally scaling each term. This feature makes geometric sequences predictable and allows us to easily determine any term if we know one term and the common ratio.

Exploring the nth Term Formula

The nth term of a geometric sequence can be calculated using the formula: \[ T_n = ar^{(n-1)} \] where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. Utilizing this formula allows for generating or identifying specific terms within your sequence with known values of \( a \) and \( r \). In our problem, we already established the ratio \( r = \frac{1}{3} \) and we know the 6th term, \( T6 = 120 \). To find the initial term, we incorporate these into the formula:

• Since \( T6 = ar^{(5)} = 120 \), solving for \( a \) requires rearranging this to \( a = \frac{120}{(\frac{1}{3})^{5}} \).

This formula is foundational in sequence calculations as it allows not only for resolving individual terms but also understanding the growth and patterns inherent within the sequence.

Conducting Sequence Calculations

Calculating terms in a geometric sequence involves applying the common ratio and harnessing the nth term formula to find missing elements. The sequence calculation primarily focuses on unraveling unknowns when certain sequence terms are given. In the exercise, with the 6th term known, you aimed to find the first term. Starting with the relationship \( ar^{(5)} = 120 \), and knowing \( r = \frac{1}{3} \), allowed us to determine the first term by solving:

• \( a = \frac{120}{(\frac{1}{3})^5} \)
• \( a = 120 \times 243 \)
• \( a = 29160 \)

Thus, the first term is \( 29160 \). Sequence calculations are essential as they help reconstruct sequences and solve for unknowns using logical steps and known parameters. Each term's position solidifies the sequence's framework, highlighting geometric sequences' potential for precision and order.