For the A.P., `a_(1),a_(2),a_(3)`,…………… if `(a_(4))/(a_(7))=2/3`, find `(a_(6))/(a_(8))`

Answer

To solve the problem, we need to find the ratio \( \frac{a_6}{a_8} \) given that \( \frac{a_4}{a_7} = \frac{2}{3} \).

### Step-by-Step Solution:

1. **Understanding the terms of the A.P.**:
In an arithmetic progression (A.P.), the \( n \)-th term can be expressed as:
\[ a_n = a + (n-1)d \]
where \( a \) is the first term and \( d \) is the common difference.

2. **Expressing \( a_4 \) and \( a_7 \)**:
Using the formula for the \( n \)-th term:
\[ a_4 = a + (4-1)d = a + 3d \]
\[ a_7 = a + (7-1)d = a + 6d \]

3. **Setting up the equation**:
We know from the problem statement that:
\[ \frac{a_4}{a_7} = \frac{2}{3} \]
Substituting the expressions for \( a_4 \) and \( a_7 \):
\[ \frac{a + 3d}{a + 6d} = \frac{2}{3} \]

4. **Cross-multiplying to eliminate the fraction**:
Cross-multiplying gives:
\[ 3(a + 3d) = 2(a + 6d) \]

5. **Expanding both sides**:
Expanding the equation:
\[ 3a + 9d = 2a + 12d \]

6. **Rearranging the equation**:
Bringing all terms involving \( a \) to one side and terms involving \( d \) to the other side:
\[ 3a - 2a = 12d - 9d \]
This simplifies to:
\[ a = 3d \]

7. **Finding \( a_6 \) and \( a_8 \)**:
Now we need to find \( a_6 \) and \( a_8 \):
\[ a_6 = a + (6-1)d = a + 5d \]
\[ a_8 = a + (8-1)d = a + 7d \]

8. **Substituting \( a = 3d \)**:
Substitute \( a = 3d \) into the expressions for \( a_6 \) and \( a_8 \):
\[ a_6 = 3d + 5d = 8d \]
\[ a_8 = 3d + 7d = 10d \]

9. **Finding the ratio \( \frac{a_6}{a_8} \)**:
Now we can find the ratio:
\[ \frac{a_6}{a_8} = \frac{8d}{10d} = \frac{8}{10} = \frac{4}{5} \]

### Final Answer:
Thus, the required ratio \( \frac{a_6}{a_8} \) is:
\[ \frac{4}{5} \]