Let the roots of the quadratic equation x^2 - 14x + k = 0 be the 2^ nd and 6^ th terms of an increasing arithmetic progression. If the sum of the first 9 terms of this arithmetic progression is 81 , then the value of k is equal to

Step-by-step solution

Let the first term of the arithmetic progression be a and the common difference be d . Given that the sum of the first 9 terms is 81 : S₉ = 9 2 (2a + 8d) = 81 2a + 8d = 18 a + 4d = 9 (1) The roots of the equation x^2 - 14x + k = 0 are the 2^ nd and 6^ th terms of the AP. Sum of the roots = 14 T₂ + T₆ = 14 (a + d) + (a + 5d) = 14 2a + 6d = 14 a + 3d = 7 (2) Subtracting equation (2) from (1): d = 2 Substituting d = 2 in equation (2): a + 6 = 7 a = 1 The roots of the quadratic equation are: T₂ = a + d = 1 + 2 = 3 T₆ =