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

Question

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

UpStudy AI · Accepted Answer

Let's determine the first three terms and the general term $$ T_h $$ of the geometric sequence.

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

### Steps to Find the Common Ratio ($$ r $$):

1. **Express the Second and Third Terms in Terms of $$ r $$:**
   - **Second term**: $$ a \cdot r = 3r $$
   - **Third term**: $$ a \cdot r^2 = 3r^2 $$

2. **Set Up the Equation Based on the Given Sum:**
   $$
   3r + 3r^2 = 60
   $$

3. **Simplify the Equation:**
   $$
   r + r^2 = 20 \quad \Rightarrow \quad r^2 + r - 20 = 0
   $$

4. **Solve the Quadratic Equation for $$ r $$:**
   $$
   r = \frac{-1 \pm \sqrt{1 + 80}}{2} = \frac{-1 \pm 9}{2}
   $$
   - **First solution**: 
     $$
     r = \frac{-1 + 9}{2} = 4
     $$
   - **Second solution**:
     $$
     r = \frac{-1 - 9}{2} = -5
     $$

### Determining the Sequences:

1. **When $$ r = 4 $$:**
   - **First term**: 3
   - **Second term**: $$ 3 \times 4 = 12 $$
   - **Third term**: $$ 3 \times 4^2 = 48 $$
   - **General term ($$ T_h $$)**:
     $$
     T_h = 3 \times 4^{h-1}
     $$

2. **When $$ r = -5 $$:**
   - **First term**: 3
   - **Second term**: $$ 3 \times (-5) = -15 $$
   - **Third term**: $$ 3 \times (-5)^2 = 75 $$
   - **General term ($$ T_h $$)**:
     $$
     T_h = 3 \times (-5)^{h-1}
     $$

### **Final Answer:**

There are two possible geometric sequences:

1. **With common ratio $$ r = 4 $$:**
   - **First three terms**: 3, 12, 48
   - **General term**: $$ T_h = 3 \times 4^{\,h-1} $$

2. **With common ratio $$ r = -5 $$:**
   - **First three terms**: 3, –15, 75
   - **General term**: $$ T_h = 3 \times (-5)^{\,h-1} $$
The first three terms are 3, 12, and 48. The general term is $$ T_h = 3 \times 4^{h-1} $$.

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

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