If the `10^(th)` term of an AP is 0 , then find the ratio of the `27^(th)` term and the `15^(th)` term of the AP.

Options

A. `1:1`
B. `17:5`
C. `1:3`
D. `3:1`

Answer

To solve the problem, we need to find the ratio of the 27th term and the 15th term of an arithmetic progression (AP) given that the 10th term is 0.

### Step-by-Step Solution:

1. **Understanding the nth term of an AP**:
The nth term of an AP can be expressed as:
\[ a_n = a + (n - 1) \cdot d \]
where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

2. **Finding the 10th term**:
Given that the 10th term \( a_{10} = 0 \):
\[ a_{10} = a + (10 - 1) \cdot d = a + 9d = 0 \]
Rearranging this gives:
\[ a + 9d = 0 \quad \Rightarrow \quad a = -9d \quad \text{(Equation 1)} \]

3. **Finding the 27th term**:
Now, we calculate the 27th term \( a_{27} \):
\[ a_{27} = a + (27 - 1) \cdot d = a + 26d \]
Substituting \( a \) from Equation 1:
\[ a_{27} = -9d + 26d = 17d \]

4. **Finding the 15th term**:
Next, we calculate the 15th term \( a_{15} \):
\[ a_{15} = a + (15 - 1) \cdot d = a + 14d \]
Again substituting \( a \) from Equation 1:
\[ a_{15} = -9d + 14d = 5d \]

5. **Finding the ratio of the 27th term to the 15th term**:
Now, we find the ratio \( \frac{a_{27}}{a_{15}} \):
\[ \frac{a_{27}}{a_{15}} = \frac{17d}{5d} \]
The \( d \) cancels out:
\[ \frac{a_{27}}{a_{15}} = \frac{17}{5} \]

### Final Answer:
The ratio of the 27th term to the 15th term of the AP is:
\[ \frac{17}{5} \]