Find five numbers in A.P. whose sum is 25 and the sum of whose squares is 135.

Sure, let's solve this step by step.

Step 1: Let's assume the five numbers in A.P. are a-d, a, a+d, a+2d, a+3d.

Step 2: According to the problem, the sum of these numbers is 25. So, we can write the equation as: (a-d) + a + (a+d) + (a+2d) + (a+3d) = 25 Solving this, we get 5a + 5d = 25 So, a + d = 5 ---- (Equation 1)

Step 3: It's also given that the sum of the squares of these numbers is 135. So, we can write the equation as: (a-d)^2 + a^2 + (a+d)^2 + (a+2d)^2 + (a+3d)^2 = 135 Solving this, we get 5a^2 + 10ad + 10d^2 = 135 So, a^2 + 2ad + 2d^2 = 27 ---- (Equation 2)

Step 4: Now, we have two equations (Equation 1 and Equation 2). We can solve these equations simultaneously to find the values of a and d.

Step 5: Substituting Equation 1 (a + d = 5) into Equation 2, we get: 5^2 = 27 This gives us d^2 = 2, so d = sqrt(2) or d = -sqrt(2)

Step 6: Substituting d = sqrt(2) into Equation 1, we get: a = 5 - sqrt(2)

Step 7: So, the five numbers in A.P. are: a-d = 5 - sqrt(2) - sqrt(2) = 5 - 2sqrt(2) a = 5 - sqrt(2) a+d = 5 - sqrt(2) + sqrt(2) = 5 a+2d = 5 + sqrt(2) a+3d = 5 + 2sqrt(2)

So, the five numbers are 5 - 2sqrt(2), 5 - sqrt(2), 5, 5 + sqrt(2), 5 + 2sqrt(2).