In an arithmetic progression the term is 25,nth term is -17 and the sum of n terms is 132.find n and the common difference

You have a typo

Which term is 25 ?

To find the value of 'n' and the common difference in the arithmetic progression, we can use the given information about the terms and their sum.

1. Let the first term of the arithmetic progression be 'a' and the common difference be 'd'.

2. We are given that the 25th term is 25. So we can write the equation:
a + (25-1)d = 25 [since the nth term of an arithmetic progression is given by a + (n-1)d]

3. We are also given that the nth term is -17. So we can write another equation:
a + (n-1)d = -17

4. Finally, we are given that the sum of 'n' terms is 132. The sum of 'n' terms of an arithmetic progression is given by:
Sum = (n/2)(2a + (n-1)d)
Plugging in the values, we get:
132 = (n/2)(2a + (n-1)d)

Now we have three equations with three variables (a, d, and n). We can solve these equations simultaneously to find their values.

Solving the equations:
Equation 1: a + 24d = 25 ----(1)
Equation 2: a + (n-1)d = -17 ----(2)
Equation 3: 132 = (n/2)(2a + (n-1)d) ----(3)

First, we can subtract the Equation (2) from the Equation (1) to eliminate the 'a' term:
24d - (n-1)d = 25 - (-17)
23d - nd + d = 42
(24-d)n = 42

Next, we can substitute the value of 'n' from Equation (4) into Equation (3):
132 = ((24-d)/2)(2a + ((24-d)-1)d)

Now we have two equations with two variables (a and d). We can solve these equations simultaneously to find their values.