Exams › GATE › Engineering Mathematics

Which one of the following is a closed form expression for the generating function of the sequence {aₙ}, where aₙ = 2n + 3 for all n = 0, 1, 2,... ?

1. 3/(1 - x)²
2. 3x/(1 - x)²
3. (2 - x)/(1 - x)²
4. (3 - x)/(1 - x)²

Correct answer: (3 - x)/(1 - x)²

Solution

The correct option represents the generating function for the sequence defined by aₙ = 2n + 3. It combines the contributions from the linear term (2n) and the constant term (3) in a way that aligns with the properties of generating functions, specifically using the formula for the generating function of a linear sequence.

Related GATE Engineering Mathematics questions

• The sum of the following infinite series is: 1/1! + 1/2! + 1/3! + 1/4! + 1/5! +...
• Which one of the following options is the correct Fourier series of the periodic function f(x) described below: f(x) = { 0 if −2 < x < −1 { 2k if −1 < x < 1; period = 4 { 0 if 1 < x < 2
• The value of lim_(x→∞) (x − √(x² + x)) is equal to
• The following inequality is true for all x close to 0. 2 − x²/3 < x sin x/(1 − cos x) < 2 What is the value of lim(x→0) x sin x/(1 − cos x) ?
• A closed thin-walled tube has thickness, t, mean enclosed area within the boundary of the centreline of tube’s thickness, Aₘ, and shear stress, τ. Torsional moment of resistance, T, of the section would be
• Consider the hemi-spherical tank of radius 13 m as shown in the figure (not drawn to scale). What is the volume of water (in m³) when the depth of water at the centre of the tank is 6 m?

⚔️ Practice GATE Engineering Mathematics free + battle 1v1 →