If `19^(th)` term of a non-zero A.P. is zero, then (`49^(th)` term) : (`29^(th)` term) is

Options

A. `3 : 1`
B. `4 : 1`
C. `2 : 1`
D. `1 : 3`

Answer

To solve the problem, we need to find the ratio of the 49th term to the 29th term of an arithmetic progression (A.P.) given that the 19th term is zero.

### Step-by-Step Solution:

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

2. **Finding the 19th Term**:
Given that the 19th term is zero:
\[ a_{19} = a + 18d = 0 \]
From this, we can express \( a \) in terms of \( d \):
\[ a = -18d \]

3. **Finding the 49th Term**:
Now, we can find the 49th term:
\[ a_{49} = a + 48d \]
Substituting the value of \( a \):
\[ a_{49} = -18d + 48d = 30d \]

4. **Finding the 29th Term**:
Next, we find the 29th term:
\[ a_{29} = a + 28d \]
Again, substituting the value of \( a \):
\[ a_{29} = -18d + 28d = 10d \]

5. **Finding the Ratio**:
Now we can find the ratio of the 49th term to the 29th term:
\[ \frac{a_{49}}{a_{29}} = \frac{30d}{10d} = 3 \]

Thus, the ratio of the 49th term to the 29th term is:
\[ \frac{a_{49}}{a_{29}} = 3 \]

### Final Answer:
The ratio of the 49th term to the 29th term is \( 3:1 \).

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