The sum of first 6terms of an A.P.is 6.The product of second term and fifth term is -80.Find the terms of A.P.
To solve this problem, we need to find the common difference (d) of the arithmetic progression (A.P.) first.
Let's assume that the first term of the A.P. is a, and the common difference is d.
The sum of the first 6 terms of an A.P. can be expressed as:
Sum = (n/2) * (2a + (n-1)d)
where n is the number of terms.
Given that the sum of the first 6 terms is 6, we have:
6 = (6/2) * (2a + 5d)
6 = 3 * (2a + 5d)
2a + 5d = 2 ----------(1)
The product of the second term and the fifth term is -80. We can express this as:
a + d * a + 4d = -80
a + 4ad + 4d^2 = -80
a + 4d(a + d) = -80
a + 4d(a + 4d - 3d) = -80
a + 4d(a + 4d) - 12d^2 = -80
a + 4a^2 + 16ad - 12d^2 = -80
4a^2 + (16d)a + (-12d^2) = -80
4a^2 + 16ad - 12d^2 + 80 = 0
a^2 + 4ad - 3d^2 + 20 = 0
(a + 5d)(a - 3d) + 20 = 0
(a + 5d)(a - 3d) = -20 ----------(2)
We have two equations (1) and (2) with two variables, a and d. We can solve them simultaneously to find the values of a and d.
From equation (2), we have two cases for the factors (a + 5d) and (a - 3d), which can form various combinations of products that multiply to -20.
Case 1:
(a + 5d) = 1
(a - 3d) = -20
Case 2:
(a + 5d) = -1
(a - 3d) = 20
Let's solve these simultaneously:
Case 1:
(a + 5d) = 1 ------------(3)
(a - 3d) = -20 ------------(4)
From equation (3), we can write:
a = 1 - 5d
Substitute the value of a in equation (4):
1 - 5d - 3d = -20
-8d = -21
d = 21/8
d = 2.625
Substitute this value of d in equation (3) to get the value of a:
a = 1 - 5d
a = 1 - 5(2.625)
a = 1 - 13.125
a = -12.125
The terms of the A.P. are:
a = -12.125
d = 2.625
So, the first term of the A.P. is -12.125, and the common difference is 2.625.
(6/2)(2a+5d) = 6
(a+d)(a+4d) = -80
or,
2a+5d = 2
a^2+5ad+4d^2 = -80
Now just solve for a and d. One sequence is
-14, -8, -2, 4, 10, 16
sum = 6
(-8)(10) = -80
There is another value for a as well