The first term of a geometric sequence is 3 and the sum of the second and third terms is 60.
Determine the first three terms and the general term .

Question

The first term of a geometric sequence is 3 and the sum of the second and third terms is 60. Determine the first three terms and the general term $$ T_{n} $$.

UpStudy AI · Accepted Answer

Let's solve the problem step by step.

### Given:
- **First term ($$ a $$)** of the geometric sequence: $$ 3 $$
- **Sum of the second and third terms**: $$ 60 $$

### Let:
- **Common ratio**: $$ r $$

### Find:
1. The **first three terms** of the sequence.
2. The **general term** $$ T_n $$.

### Step 1: Express the Terms
- **First term**: $$ T_1 = a = 3 $$
- **Second term**: $$ T_2 = ar = 3r $$
- **Third term**: $$ T_3 = ar^2 = 3r^2 $$

### Step 2: Set Up the Equation
Given that the sum of the second and third terms is $$ 60 $$:
$$
T_2 + T_3 = 60 \
3r + 3r^2 = 60
$$

### Step 3: Solve for $$ r $$
Divide the entire equation by $$ 3 $$:
$$
r + r^2 = 20 \
r^2 + r - 20 = 0
$$

Solve the quadratic equation:
$$
r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \
a = 1, \, b = 1, \, c = -20 \
r = \frac{-1 \pm \sqrt{1 + 80}}{2} \
r = \frac{-1 \pm 9}{2}
$$

**Possible values for $$ r $$:**
1. $$ r = \frac{-1 + 9}{2} = 4 $$
2. $$ r = \frac{-1 - 9}{2} = -5 $$

### Step 4: Determine the First Three Terms

**Case 1:** $$ r = 4 $$
$$
T_1 = 3 \
T_2 = 3 \times 4 = 12 \
T_3 = 3 \times 4^2 = 48 \
$$
**First Three Terms:** $$ 3, \, 12, \, 48 $$

**Case 2:** $$ r = -5 $$
$$
T_1 = 3 \
T_2 = 3 \times (-5) = -15 \
T_3 = 3 \times (-5)^2 = 75 \
$$
**First Three Terms:** $$ 3, \, -15, \, 75 $$

### Step 5: Write the General Term $$ T_n $$
The general term of a geometric sequence is given by:
$$
T_n = ar^{n-1}
$$

**For $$ r = 4 $$:**
$$
T_n = 3 \times 4^{n-1}
$$

**For $$ r = -5 $$:**
$$
T_n = 3 \times (-5)^{n-1}
$$

### **Summary**

- **First Three Terms:**
  - If $$ r = 4 $$: $$ 3, \, 12, \, 48 $$
  - If $$ r = -5 $$: $$ 3, \, -15, \, 75 $$

- **General Term $$ T_n $$:**
  - If $$ r = 4 $$: $$ T_n = 3 \times 4^{n-1} $$
  - If $$ r = -5 $$: $$ T_n = 3 \times (-5)^{n-1} $$
The first three terms are 3, 12, and 48 when the common ratio is 4, or 3, -15, and 75 when the common ratio is -5. The general term is $$ T_n = 3 \times 4^{n-1} $$ for $$ r = 4 $$ or $$ T_n = 3 \times (-5)^{n-1} $$ for $$ r = -5 $$.

The first term of a geometric sequence is 3 and the sum of the second and third terms is 60.
Determine the first three terms and the general term .

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Algebra Questions

The first term of a geometric sequence is 3 and the sum of the second and third terms is 60. Determine the first three terms and the general term .

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Algebra Questions

The first term of a geometric sequence is 3 and the sum of the second and third terms is 60.
Determine the first three terms and the general term .