The ratio of the 7th to the 3rd terms of an A.P. is 12:5, find the ratio of the 13th to 4th term :

Options

A. `13:7`
B. `4:3`
C. `10:3`
D. none of these

Answer

To solve the problem step by step, we will use the properties of an Arithmetic Progression (A.P.).

### Step 1: Understand the terms in A.P.
In an A.P., the nth term (Tn) can be expressed as:
\[ T_n = a + (n - 1)d \]
where \( a \) is the first term and \( d \) is the common difference.

### Step 2: Write the expressions for the 7th and 3rd terms.
Using the formula for the nth term:
- The 7th term (T7) is:
\[ T_7 = a + 6d \]
- The 3rd term (T3) is:
\[ T_3 = a + 2d \]

### Step 3: Set up the ratio of the 7th to the 3rd term.
According to the problem, the ratio of the 7th term to the 3rd term is given as:
\[ \frac{T_7}{T_3} = \frac{12}{5} \]
Substituting the expressions for T7 and T3:
\[ \frac{a + 6d}{a + 2d} = \frac{12}{5} \]

### Step 4: Cross-multiply to eliminate the fraction.
Cross-multiplying gives:
\[ 5(a + 6d) = 12(a + 2d) \]
Expanding both sides:
\[ 5a + 30d = 12a + 24d \]

### Step 5: Rearrange the equation to isolate terms.
Rearranging the equation:
\[ 5a - 12a = 24d - 30d \]
This simplifies to:
\[ -7a = -6d \]

### Step 6: Solve for the relationship between a and d.
Dividing both sides by -1:
\[ 7a = 6d \]
Thus, we can express \( a \) in terms of \( d \):
\[ a = \frac{6}{7}d \]

### Step 7: Find the expressions for the 13th and 4th terms.
Now we need to find the 13th term (T13) and the 4th term (T4):
- The 13th term (T13) is:
\[ T_{13} = a + 12d = \frac{6}{7}d + 12d = \frac{6}{7}d + \frac{84}{7}d = \frac{90}{7}d \]
- The 4th term (T4) is:
\[ T_4 = a + 3d = \frac{6}{7}d + 3d = \frac{6}{7}d + \frac{21}{7}d = \frac{27}{7}d \]

### Step 8: Set up the ratio of the 13th to the 4th term.
Now, we find the ratio of T13 to T4:
\[ \frac{T_{13}}{T_4} = \frac{\frac{90}{7}d}{\frac{27}{7}d} \]
The \( d \) and \( \frac{1}{7} \) cancel out:
\[ \frac{T_{13}}{T_4} = \frac{90}{27} \]

### Step 9: Simplify the ratio.
Simplifying \( \frac{90}{27} \) gives:
\[ \frac{90 \div 9}{27 \div 9} = \frac{10}{3} \]

### Final Answer:
Thus, the ratio of the 13th term to the 4th term is:
\[ \frac{T_{13}}{T_4} = \frac{10}{3} \]

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