If a1,a2,a3 represents first, second, third terms in A.P. if the first term is 2 and (a1 + a2 ) a3 is minimum then
the common difference is ___?
Answer is -5/2
To find the common difference in an arithmetic progression (A.P.) where the sum of the product of the first two terms and the third term is minimum, we can use the following steps:
Step 1: Express the terms in A.P.
Given: a1 = 2
Since the terms are in A.P., we have:
a1 = 2
a2 = a1 + d
a3 = a1 + 2d
Step 2: Calculate the expression to be minimized.
The given condition states that (a1 + a2) * a3 should be minimum. So, we have:
Expression = (a1 + a2) * a3
Expression = (2 + a1 + d) * (a1 + 2d)
Expression = (2 + 2 + d) * (2 + 2d)
Expression = (4 + d) * (2 + 2d)
Expression = 8 + 12d + 4d + 2d^2
Expression = 2d^2 + 16d + 8
Step 3: Find the minimum point.
To minimize the expression, we need to find the minimum point of the quadratic equation 2d^2 + 16d + 8. This can be done by differentiating the equation and setting it equal to zero.
Differentiating the equation:
d(Expression)/dd = 4d + 16
Setting d(Expression)/dd = 0:
4d + 16 = 0
4d = -16
d = -16/4
d = -4
Step 4: Determine the common difference.
Since we found that d = -4, the common difference in the given arithmetic progression is -4.
Therefore, the common difference is -4.
To find the common difference in an arithmetic progression (A.P.), we need to recognize that the sum of any three terms in an A.P. can be written as:
(a + (a + d) + (a + 2d)) = (3a + 3d),
where 'a' is the first term and 'd' is the common difference.
In this case, we are given that the first term, a1, is 2, so a = 2.
We also know that (a1 + a2) * a3 is minimum. Therefore, we need to find the minimum value of (3a + 3d) * (a + 2d).
Expanding and simplifying the expression, we have:
(3a^2 + 9ad + 6d^2).
To find the minimum, we can analyze the quadratic expression (3a^2 + 9ad + 6d^2) with respect to 'd'.
The minimum value of a quadratic expression (ax^2 + bx + c) occurs at x = -b/2a.
For our expression (3a^2 + 9ad + 6d^2), the coefficient of 'd^2' is 6, so a = 6.
Therefore, we can plug in our values back into (3a^2 + 9ad + 6d^2) to find the minimum:
(3(6)^2 + 9(6)(2) + 6(2)^2) = 216 + 108 + 24 = 348.
As we have found the minimum value of the given expression, the common difference (d) is the square root of the coefficient of d^2.
Solving for d, we get the common difference:
√(6) ≈ 2.449.
Therefore, the common difference is approximately 2.449.