(c) Consider the geometric sequence $$ \frac{2}{9} ;-\frac{2}{3} ; 2 ; \ldots ; 1062882 $$. (1) $$ \quad $$ Determine the general term. (2) Calculate the value of the $$ 10^{ ext {th }} $$ term. (3) Which term in the sequence is equal to -486 ? (4) How many terms are there in this sequence?

Question

(c) Consider the geometric sequence $$ \frac{2}{9} ;-\frac{2}{3} ; 2 ; \ldots ; 1062882 $$. (1) $$ \quad $$ Determine the general term. (2) Calculate the value of the $$ 10^{	ext {th }} $$ term. (3) Which term in the sequence is equal to -486 ? (4) How many terms are there in this sequence?

UpStudy AI · Accepted Answer

Let's analyze the given geometric sequence step by step:

$$ \frac{2}{9},\ -\frac{2}{3},\ 2,\ \ldots,\ 1062882 $$

### (1) **Determine the General Term**

A geometric sequence has the general form:

$$ a_n = a_1 \cdot r^{n-1} $$

Where:
- $$ a_1 $$ is the first term.
- $$ r $$ is the common ratio.
- $$ a_n $$ is the $$ n^{\text{th}} $$ term.

**Finding the Common Ratio ($$ r $$):**

Given:
- First term, $$ a_1 = \frac{2}{9} $$
- Second term, $$ a_2 = -\frac{2}{3} $$

$$
r = \frac{a_2}{a_1} = \frac{-\frac{2}{3}}{\frac{2}{9}} = -\frac{2}{3} \times \frac{9}{2} = -3
$$

**General Term Formula:**

$$
a_n = \frac{2}{9} \cdot (-3)^{n-1}
$$

### (2) **Calculate the $$ 10^{\text{th}} $$ Term**

Using the general term formula:

$$
a_{10} = \frac{2}{9} \cdot (-3)^{10-1} = \frac{2}{9} \cdot (-3)^9
$$

Calculate $$ (-3)^9 $$:

$$
(-3)^9 = -19683
$$

Now, compute $$ a_{10} $$:

$$
a_{10} = \frac{2}{9} \times (-19683) = \frac{-39366}{9} = -4374
$$

**So, the $$ 10^{\text{th}} $$ term is** $$ \boxed{-4374} $$.

### (3) **Which Term is Equal to -486?**

We need to find $$ n $$ such that:

$$
a_n = \frac{2}{9} \cdot (-3)^{n-1} = -486
$$

Solving for $$ (-3)^{n-1} $$:

$$
(-3)^{n-1} = -486 \times \frac{9}{2} = -2187
$$

Recognize that:

$$
-2187 = (-3)^7
$$

Thus:

$$
n - 1 = 7 \quad \Rightarrow \quad n = 8
$$

**So, the term equal to -486 is the** $$ \boxed{8^{\text{th}}} $$ **term**.

### (4) **How Many Terms are in the Sequence?**

The last term is 1,062,882. We need to find $$ n $$ such that:

$$
a_n = \frac{2}{9} \cdot (-3)^{n-1} = 1062882
$$

Solving for $$ (-3)^{n-1} $$:

$$
(-3)^{n-1} = 1062882 \times \frac{9}{2} = 4782969
$$

Recognize that:

$$
4782969 = 3^{14} = (-3)^{14}
$$

Thus:

$$
n - 1 = 14 \quad \Rightarrow \quad n = 15
$$

**So, there are** $$ \boxed{15} $$ **terms in the sequence**.
**General Term:** $$ a_n = \frac{2}{9} \cdot (-3)^{n-1} $$ - **10th Term:** $$ -4374 $$ - **Term Equal to -486:** 8th term - **Total Terms:** 15

© Consider the geometric sequence .
(1) Determine the general term.
(2) Calculate the value of the term.
(3) Which term in the sequence is equal to -486 ?
(4) How many terms are there in this sequence?

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Pre Algebra Questions

© Consider the geometric sequence . (1) Determine the general term. (2) Calculate the value of the term. (3) Which term in the sequence is equal to -486 ? (4) How many terms are there in this sequence?