The sum of first 6 terms of an Arithmetic progression is 6 the product of 2nd term and 5th term is -80 . Find the terms of A.P.
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To find the terms of the arithmetic progression (A.P.), let's consider the given information step by step:
1. The sum of the first 6 terms of the A.P. is 6.
To find the sum of the first 6 terms of an A.P., we can use the formula: Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the n terms, a is the first term, and d is the common difference.
Let's use this formula to solve the first part of the problem:
6 = (6/2)(2a + (6-1)d)
6 = 3(2a + 5d)
2a + 5d = 2 -- Equation 1
2. The product of the 2nd term and the 5th term is -80.
Let's denote the 2nd term as a2 and the 5th term as a5. In an A.P., the nth term can be written as an = a + (n-1)d.
So, we have the following equation for the 2nd term and 5th term of the A.P.:
a2 * a5 = -80
(a + d) * (a + 4d) = -80
a^2 + 5ad + 4d^2 = -80 -- Equation 2
Now, we have two equations (Equation 1 and Equation 2) with two variables (a and d). We can solve these equations simultaneously to find the values of a and d.
Let's substitute the value of a from Equation 1 into Equation 2 and solve for d:
From Equation 1, we have: 2a = 2 - 5d
Substituting this value of 2a into Equation 2:
(2 - 5d)^2 + 5(2 - 5d)d + 4d^2 = -80
4 - 20d + 25d^2 + 10d - 25d^2 + 4d^2 = -80
4 - 10d + 4d^2 = -80
4d^2 - 10d + 84 = 0
Now, we can solve this quadratic equation to find the values of d.
Using the quadratic formula: d = (-b ± √(b^2 - 4ac))/(2a)
In our equation, a = 4, b = -10, and c = 84.
Calculating the values of d using the quadratic formula, we get two possible values: d = -3 and d = 7.
Now, we can substitute these values of d back into Equation 1 to find the corresponding values of a:
For d = -3:
2a + 5(-3) = 2
2a - 15 = 2
2a = 17
a = 8.5
For d = 7:
2a + 5(7) = 2
2a + 35 = 2
2a = -33
a = -16.5
Hence, the two arithmetic progressions are:
A.P. with a = 8.5 and d = -3
A.P. with a = -16.5 and d = 7