If the ordinary generating function of a sequence is:
then a3-a0 is equal to:
(A) 8
(B) 10
(C) 15
(D) 20
Answer: (C)
Explanation: Given, ordinary generating function of an infinite sequence is:
Since, (1−z)−3 = 1+(3C1).z + (4C2).z2 + (5C3).z3 + … ∞
So, (1+z)(1−z)−3 = (1+z)∗(1+(3C1).z + (4C2).z2 + (5C3).z3 + … ∞)
Therefore, coefficient of z0 = a0 = 1,
and a0 = coefficient of z3 = a3 = 4C2 + 5C3 = 6 + 10 = 16.
Hence, a3 – a0 = 16 – 1 = 15
Alternative way –
(z(1+z))/(1-z)3 = ∑r2.zr => (1+z)/(1-z)3 = 1/z(0.z0 + 1.z1 + 4.z2 + 9.z3 + 16.z4 + ... ) => the sequence generated = 1.z0, 4.z1, 9.z2, 16.z3, ... => a0 = 1; and a3 = 16 => a3 - a0 = 16 - 1 = 15
This explanation is contributed by Krit Verma.