Let the sum of the first n terms of a non-constant A.P., a1, a2, a3, ..... be 50n + \frac{n \left(\right. n - 7 \left.\right)}{2} A, where A is a constant. If d is the common difference of this A.P., then the ordered pair (d, a50) is equal to

Engineering

Mathematics

Arithmetic Progression and Its Properties

Question

Let the sum of the first n terms of a non-constant A.P., a₁, a₂, a₃, ..... be 50n + $\frac{n (n - 7)}{2}$ A, where A is a constant. If d is the common difference of this A.P., then the ordered pair (d, a₅₀) is equal to

(50, 50 + 45A)

(50, 50 + 46A)

(A, 50 + 45A)

(A, 50 + 46A)

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$S_{n} = 50 n + \frac{n (n - 7)}{2} A$

T_n = S_n + S_n-1

_{$= 50 n + \frac{n (n - 7)}{2} A - 50 (n - 1) - \frac{(n - 1) (n - 8)}{2} A$}

_{$= 50 n + \frac{A}{2} [n^{2} - 7 n - n^{2} + 9 n - 8]$}

= 50 + A (n – 4)

d = T_n – T_n–1

= 50 + A (n – 4) – 50 – A (n – 5)

= A

T₅₀ = 50 + 46A

(d, A₅₀) = (A, 50 + 46A)