Theory question 110
Sequences and SeriesGiven that 8Cx, 7Cx and (7/6)(6Cx) form the first three consecutive terms of an exponential sequence (G.P.), find the: (a) value of x; (b) common ratio of the sequence; (c) sum of the first ten terms of the sequence.
Show the model answer
Model answer
(a) (⁷Cₓ)² = ⁸Cₓ × (7/6)⁶Cₓ. Using ⁸Cₓ = 8/(8 − x) × ⁷Cₓ and ⁶Cₓ = (7 − x)/7 × ⁷Cₓ:
1 = 8(7 − x)/(6(8 − x)), so 48 − 6x = 56 − 8x and x = 4.
(b) Terms: ⁸C₄ = 70, ⁷C₄ = 35, (7/6)(15) = 17.5; r = 1/2.
(c) S₁₀ = 70(1 − (½)¹⁰)/(1 − ½) = 140 × 1023/1024 ≈ 139.86.
Explanation
For a G.P., (middle term)² = first × third.