Question

If sum of series (x + ka) + (x2 + (k - 2}a) + (x3 + (k - 4)a) + ... 9 terms is {x10 - x -45a(x-1)} / (x-1) then value of k is :

Answer 1

Sum of series (x + ka) + (x² + (k - 2)a) + (x³ + (k - 4)a) + ...... 9 terms is (x¹⁰ - x - 45a(x - 1)}/(x - 1)

we have to find the value of k

solution : (x + ka) + (x² + (k - 2)a) + (x³ + (k - 4)a) + ...... 9 terms

= [x + x² + x³ + .... + 9 terms ] + a[k + (k - 2) + (k - 4) + ... + 9 terms ]

using formula, Sn = a(rⁿ - 1)/(r - 1)

= x(x^9 - 1)/(x - 1) + a[(k + k + k + ... 9 terms) - (0 + 2 + 4 + 6 + .... 9 terms)]

= x(x^9 - 1)/(x - 1) + a[9k - 9/2 (2 × 0 + (9 - 1) × 2)]

= x(x^9 - 1)/(x - 1) + a[9k - 72 ]

= {(x¹⁰ - x) + (9ka -72a)(x - 1)}/(x - 1)

on comparing to (x¹⁰ - x - 45a(x - 1)}/(x - 1)

we get, (9ka - 72a) = - 45a

⇒9ka = 72a - 45a = 27a

⇒k = 3

Therefore the value of k = 3