3rd term = a + 2d = 8 

Solution:

Given that the 3rd term of the arithmetic progression (A.P.) is 8 and the 7th term is 9 more than the 4th term. We need to find the sum of the first 20 terms of the A.P.

Step 1: Finding the Common Difference (d)
In an arithmetic progression, the difference between any two consecutive terms is constant. Let's find the common difference (d) using the given information.

The 3rd term (a3) is given as 8, and it can be represented as:
a3 = a1 + 2d (where a1 is the first term)

Substituting the given values:
8 = a1 + 2d

Similarly, the 7th term (a7) is given as 9 more than the 4th term (a4), which can be represented as:
a7 = a4 + 3d

Substituting the given values:
a4 + 3d = a1 + 6d + 9

Since we have two equations with two variables (a1 and d), we can solve them simultaneously to find their values.

Step 2: Solving the Equations
Using the equations derived in Step 1, we can solve for a1 and d.

Equation 1: 8 = a1 + 2d
Equation 2: a4 + 3d = a1 + 6d + 9

Rearranging Equation 2:
a1 - a4 = -3d - 9 ...(3)

Substituting Equation 1 into Equation 3:
8 - a4 = -3d - 9

Rearranging the equation:
a4 - 17 = 3d

Step 3: Finding the Common Difference (d)
From the above equation, we can find the value of d by comparing the coefficients of d on both sides.

Comparing coefficients:
3d = a4 - 17

Therefore, the common difference (d) is given by:
d = (a4 - 17)/3

Step 4: Finding the First Term (a1)
Now, we can substitute the value of d into Equation 1 to find the first term (a1).

8 = a1 + 2d

Substituting the value of d:
8 = a1 + 2[(a4 - 17)/3]

Simplifying the equation:
8 = a1 + (2/3)(a4 - 17)

Multiplying through by 3 to eliminate the fraction:
24 = 3a1 + 2(a4 - 17)

Expanding the equation:
24 = 3a1 + 2a4 - 34

Combining like terms:
3a1 + 2a4 = 58 ...(4)

Step 5: Finding the Sum of the First 20 Terms
To find the sum of the first 20 terms of the A.P., we can use the formula for the sum of an arithmetic series:

S20 = (n/2)(2a1 + (n-1)d)

Substituting the values we have found:
S20 = (20/2)(2a1 +