Given Information
- The sum of the first four terms of an Arithmetic Progression (AP) is 8.
- The product of the extreme terms and the middle terms is in the ratio of 5:6.
Let’s Define the Terms
- Let the first term be \( a \) and the common difference be \( d \).
- The first four terms of the AP can be represented as:
- \( a, a + d, a + 2d, a + 3d \)
Sum of the First Four Terms
- The sum can be calculated as:
\[
S = a + (a + d) + (a + 2d) + (a + 3d) = 4a + 6d
\]
- Given that \( S = 8 \):
\[
4a + 6d = 8 \quad (1)
\]
Product of Terms
- The extreme terms are \( a \) and \( a + 3d \).
- The middle terms are \( a + d \) and \( a + 2d \).
- The products are:
\[
\text{Extreme terms product} = a(a + 3d) = a^2 + 3ad
\]
\[
\text{Middle terms product} = (a + d)(a + 2d) = a^2 + 3ad + 2d^2
\]
- Given the ratio:
\[
\frac{a(a + 3d)}{(a + d)(a + 2d)} = \frac{5}{6} \quad (2)
\]
Solve the Equations
1. From equation (1):
\[
2a + 3d = 4 \quad (3)
\]
2. Substitute \( a \) from equation (3) into equation (2) and simplify.
Final Calculation
- Solving these equations will yield values for \( a \) and \( d \).
- After finding \( a \) and \( d \), calculate the first four terms as:
\[
a, a+d, a+2d, a+3d
\]
Conclusion
- After substitutions and calculations, you will find the first four terms of the AP that satisfy both conditions.