Problem 17

Question

Find the first five terms of the sequence and determine if it is geometric. If it is geometric, find the common ratio and express the $n$th term of the sequence in the standard form $a_{n}=a r^{n-1} .$ $$ a_{n}=2(3)^{n} $$

Step-by-Step Solution

Verified

Answer

The sequence is geometric with first five terms: 6, 18, 54, 162, 486. Common ratio: 3. Standard form: $a_n = 6 \cdot 3^{n-1}$.

1Step 1: Identify the Given Expression

We start by noting the given expression for the sequence: $ a_n = 2(3)^n $. This already resembles the geometric sequence formula, where $ a_n = ar^{n-1} $.

2Step 2: Generate the First Five Terms

Substitute $ n = 1, 2, 3, 4, 5 $ into the expression $ a_n = 2(3)^n $ to find the first five terms:1. $ a_1 = 2(3)^1 = 6 $2. $ a_2 = 2(3)^2 = 18 $3. $ a_3 = 2(3)^3 = 54 $4. $ a_4 = 2(3)^4 = 162 $5. $ a_5 = 2(3)^5 = 486 $.

3Step 3: Determine if the Sequence is Geometric

Check if the ratio between consecutive terms is constant:$ \frac{a_2}{a_1} = \frac{18}{6} = 3 $$ \frac{a_3}{a_2} = \frac{54}{18} = 3 $$ \frac{a_4}{a_3} = \frac{162}{54} = 3 $$ \frac{a_5}{a_4} = \frac{486}{162} = 3 $.Since the ratio is constant, the sequence is geometric with a common ratio $ r = 3 $.

4Step 4: Express the Sequence in Standard Form

In the standard form for a geometric sequence $ a_n = ar^{n-1} $, identify $ a = 6 $ from the first term $ a_1 = 6 $ and $ r = 3 $. The expression $ a_n $ becomes:\[ a_n = 6 \cdot 3^{n-1} \]This matches the standard form for geometric sequences.

Key Concepts

Common RatioN$n$th Term FormulaSequence TermsStep by Step Solution

Common Ratio

In a geometric sequence, the common ratio is a crucial element. It specifies how each term in the sequence relates to the previous term.
To determine the common ratio, you simply divide any term in the sequence by its preceding term. This ratio remains constant throughout a geometric sequence.
For example, in the original exercise, the common ratio can be found by dividing the second term by the first, the third by the second, and so on. Each calculation yields 3:

$ \frac{18}{6} = 3 $
$ \frac{54}{18} = 3 $
$ \frac{162}{54} = 3 $
$ \frac{486}{162} = 3 $

This constant ratio confirms the presence of a geometric sequence with a common ratio of 3.

N$n$th Term Formula

The nth term formula for a geometric sequence allows you to quickly find any term in the sequence without listing all preceding terms.
It is expressed as:\[ a_n = a \cdot r^{n-1} \]Here:

$a_n$ is the nth term
$a$ is the first term of the sequence
$r$ is the common ratio
$n$ is the position of the term in the sequence

In our example, the modified expression for the sequence in the standard form is $a_n = 6 \cdot 3^{n-1}$.
By understanding this formula, determining specific terms becomes simpler and more efficient.

Sequence Terms

In the context of geometric sequences, sequence terms are individual elements within the sequence.
They are generated by substituting values into the nth term formula.
For the given sequence $a_n = 2(3)^n$, the first five terms are calculated by plugging in consecutive values of $n$:

$a_1 = 2(3)^1 = 6$
$a_2 = 2(3)^2 = 18$
$a_3 = 2(3)^3 = 54$
$a_4 = 2(3)^4 = 162$
$a_5 = 2(3)^5 = 486$

These calculations provide a clear representation of how each subsequent term is derived by multiplying the previous term by the common ratio.

Step by Step Solution

A step by step approach helps demystify the process of analyzing and understanding geometric sequences.
By breaking down the process into understandable steps, the task is more manageable for students.Let's recap the solution provided:1. **Identify the Given Expression:** First, recognize the expression is part of a geometric sequence.2. **Generate the First Five Terms:** Calculate these by substituting different values for $n$ into the formula.3. **Determine if the Sequence is Geometric:** Verify that the ratio between consecutive terms is constant.4. **Express in Standard Form:** Reformat the nth term formula using identified values for first term and common ratio.Taking this systematic journey ensures clarity, allowing one to follow through with aligned comprehension of the underlying principles of geometric sequences.

Problem 17