The sum of an infinite geometric series is 162 and the sum of its first `n` terms is 160. If the inverse of its common ratio is an integer, then which of the following is not a possible first term? `108` b. `144` c. `160` d. none of these

Options

A. 108
B. 120
C. 144
D. 160

Explanation

To solve the problem, we need to analyze the information given about the infinite geometric series and the sum of its first n terms.

### Step-by-step Solution:

1. **Understanding the Formulas**:
- The sum of an infinite geometric series is given by the formula:
\[ S_{\infty} = \frac{a}{1 - r} \]
where \( a \) is the first term and \( r \) is the common ratio. We know that \( S_{\infty} = 162 \).
- The sum of the first \( n \) terms of a geometric series is given by:
\[ S_n = a \frac{1 - r^n}{1 - r} \]
We know that \( S_n = 160 \).

2. **Setting Up the Equations**:
- From the infinite series:
\[ \frac{a}{1 - r} = 162 \quad \text{(1)} \]
- From the sum of the first n terms:
\[ a \frac{1 - r^n}{1 - r} = 160 \quad \text{(2)} \]

3. **Dividing the Two Equations**:
- Divide equation (2) by equation (1):
\[ \frac{a \frac{1 - r^n}{1 - r}}{\frac{a}{1 - r}} = \frac{160}{162} \]
- This simplifies to:
\[ 1 - r^n = \frac{80}{81} \quad \text{(3)} \]

4. **Finding \( r^n \)**:
- Rearranging equation (3):
\[ r^n = 1 - \frac{80}{81} = \frac{1}{81} \]

5. **Finding the Common Ratio**:
- Since \( r^n = \frac{1}{81} \), we can express \( r \) in terms of \( n \):
\[ r = \left(\frac{1}{81}\right)^{\frac{1}{n}} = \frac{1}{3^{\frac{4}{n}}} \]

6. **Finding \( a \)**:
- Substitute \( r \) back into equation (1):
\[ a = 162(1 - r) = 162\left(1 - \frac{1}{3^{\frac{4}{n}}}\right) \]

7. **Finding Possible Values of \( a \)**:
- We need to check for integer values of \( \frac{1}{r} \) which can be \( 3, 9, 27, \) or \( 81 \) (since \( 81 = 3^4 \)).
- For \( \frac{1}{r} = 3 \):
\[ a = 162\left(1 - \frac{1}{3}\right) = 162 \cdot \frac{2}{3} = 108 \]
- For \( \frac{1}{r} = 9 \):
\[ a = 162\left(1 - \frac{1}{9}\right) = 162 \cdot \frac{8}{9} = 144 \]
- For \( \frac{1}{r} = 27 \):
\[ a = 162\left(1 - \frac{1}{27}\right) = 162 \cdot \frac{26}{27} = 156 \]
- For \( \frac{1}{r} = 81 \):
\[ a = 162\left(1 - \frac{1}{81}\right) = 162 \cdot \frac{80}{81} = 160 \]

8. **Conclusion**:
- The possible values of \( a \) are \( 108, 144, 160 \).
- The question asks which of the following is not a possible first term. Since all options \( 108, 144, 160 \) are possible, the answer is:
\[ \text{d. none of these} \]