Theory question 109
Sequences and SeriesGiven that nC4, nC5 and nC6 are the first 3 terms of a linear sequence (A. P.), find the: (a) values of n; (b) common differences of the sequence.
Show the model answer
Model answer
(a) 2 ⁿC₅ = ⁿC₄ + ⁿC₆. Divide by ⁿC₄: 2(n − 4)/5 = 1 + (n − 4)(n − 5)/30.
12(n − 4) = 30 + (n − 4)(n − 5), so n² − 21n + 98 = 0, (n − 7)(n − 14) = 0: n = 7 or 14.
(b) n = 7: terms 35, 21, 7, d = −14. n = 14: terms 1001, 2002, 3003, d = 1001.
Explanation
In an A.P. the middle term is the average of its neighbours.