Question

Use the method of generating functions to solve the following recurrence relations: aₙ +3aₙ₋₁−4aₙ₋₂ =0;n≥2, given aₒ =3,a₁ =−2.

aₙ₋₂−5aₙ₊₁ +6aₙ =36;n≥0; given aₒ = a₁=0
aₙ₊₂ - aₙ = 2ⁿ ;n≥0; given .aₒ =0,a₁ =1
aₙ₊₂ − 6aₙ₊₁+9yₙ=3 ;n≥0; given aₒ = 2 = a₁=9.
aₙ₊ +4aₙ +4aₙ₋₁ =n−1;n≥1, given .aₒ =0,a₁ =1.
aₙ₊₂ + aₙ=n⋅2ⁿ ;n≥0. How many integers between 1 and 300 (both inclusive) are divisible by (i) at least one of 3,5,7 ? (ii) 3 and by 5 , but not by 7 ? (iii) 5 but by neither 3 nor 7

Answer 1

To solve the given recurrence relations using generating functions, find the generating functions for the sequences, represent the sequences as power series, manipulate the series using algebraic operations, and find the coefficients of the power series to solve the recurrence relations.

Step-by-step explanation:

To solve the given recurrence relations using the method of generating functions, we need to find the generating functions for the sequences. We can represent each sequence as a power series and manipulate the series using algebraic operations to find the generating functions. By finding the coefficients of the power series, we can find the solution to the recurrence relations.

For example, to solve the first recurrence relation aₙ +3aₙ₋₁−4aₙ₋₂ =0:

Let G(x) be the generating function for the sequence aₙ. We can represent the sequence as a power series G(x) = a₀ + a₁x + a₂x² + ...

By substituting the given initial conditions a₀ = 3 and a₁ = -2, we can write G(x) = 3 - 2x + a₂x² + ...

Multiplying both sides of the recurrence relation by x³ (to shift the indices), we get:

G(x) - 3x³G(x) + 4x²G(x) = 0

Factoring out G(x) and solving for G(x), we get:

G(x) = 3x³ / (1 - 4x² + 3x³)

To find the coefficients of the power series, we can use the technique of partial fractions and write G(x) as:

G(x) = 3x³ / [(1 - x)(1 - 3x)] = A / (1 - x) + B / (1 - 3x)

Solving for A and B, we get A = 2 and B = 1. Therefore, the generating function G(x) can be written as:

G(x) = 2/(1 - x) + 1/(1 - 3x)

Using the formula for the sum of a geometric series, we can write G(x) as:

G(x) = 2(1 + x + x² + ...) + (1 + 3x + 9x² + ...)

By comparing the coefficients of the power series with the coefficients of the geometric series, we can find the sequence aₙ. In this case, aₙ can be written as:

aₙ = 2 + 1*3ⁿ

Using similar steps, you can solve the other recurrence relations as well.