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Chapter 13: Problem 21

Determine an expression for the general term of each arithmetic sequence. $$ a_{1}=2, d=5 $$

Short Answer

Expert verified
The general term is \( a_n = 10 \times \big(-\frac{1}{5}\big)^{(n-1)} \)

Step by step solution

01

Identify the first term

The first term of the geometric sequence is the initial value: \[ a_1 = 10 \]
02

Determine the common ratio

The common ratio (\r) is found by dividing the second term by the first term: \[ r = \frac{a_2}{a_1} = \frac{-2}{10} = -\frac{1}{5} \]
03

Write the general term of the geometric sequence

The general term (_t\term_n) of a geometric sequence can be expressed as: \[ a_n = a_1 \times r^{(n-1)} \] Substituting the known values: \[ a_n = 10 \times \bigg(-\frac{1}{5}\bigg)^{(n-1)} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

General Term
The general term of a geometric sequence gives you a formula to find any term in the sequence.
It is written as \(a_n\).
The formula for the general term is: \[a_n = a_1 \times r^{(n-1)}\].
Here, 'n' is the position of the term in the sequence.
This formula can easily find the nth term without having to calculate all the preceding terms.
For example, in our sequence (10, -2, \(\frac{2}{5}\), ...), the first term is 10, and the common ratio is -\(\frac{1}{5}\), so the general term is: \[a_n = 10 \times \bigg(-\frac{1}{5}\bigg)^{(n-1)}\].
By substituting different values of 'n', you can determine the values of specific terms in the sequence.
First Term
The first term of a geometric sequence is the initial value from which the sequence starts.
It is usually denoted as \(a_1\).
Finding the first term is simple, as it is typically given as the very first number in the sequence itself.
In our example sequence (10, -2, \(\frac{2}{5}\), ...), the first term is 10.
This term forms the base for subsequent calculations in the sequence.
It is important to correctly identify this term since it influences the entire sequence when calculating other values.
Common Ratio
The common ratio is the constant factor between consecutive terms of a geometric sequence.
It is typically represented by 'r'.
This ratio is found by dividing any term by the previous term.
For instance, in our example (10, -2, \(\frac{2}{5}\), ...), the common ratio can be calculated as:
  • First, identify the second term (-2) and first term (10)
  • Then divide the second term by the first, so: \( r = \frac{-2}{10} = -\frac{1}{5} \)

This common ratio helps in determining all subsequent terms using the general term formula.
A consistent common ratio is crucial for correctly defining and calculating terms in the sequence.

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