Let S denotes the infinite sum 2 + 5x + 9x^2 + 14x^3 + 20x^4 + ... , where |x| < 1 and the coefficient of x^{n - 1} is n( n + 3 )/2 , ( n = 1, 2 , . . . ) . Then S equals:

Question 27

Let S denotes the infinite sum $$2 + 5x + 9x^2 + 14x^3 + 20x^4 + ...$$ , where |x| < 1 and the coefficient of $$x^{n - 1}$$ is n( n + 3 )/2 , ( n = 1, 2 , . . . ) . Then S equals:

Solution

Let $$S = 2+5x+9x^2+....$$
$$S*x = 2x+5x^2+9x^3+...$$
$$S(1-x) = 2+3x+4x^2+...$$
$$S(1-x)*x = 2x+3x^2+4x^3+...$$
$$S(1-x)(1-x) = 2+x+x^2+x^3+... = 2+x/(1-x)$$
So, $$S = [2(1-x) + x]/(1-x)^3 => S = (2-x)/(1-x)^3$$

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CAT 2002 QA Question

Let S denotes the infinite sum $$2 + 5x + 9x^2 + 14x^3 + 20x^4 + ...$$ , where |x| < 1 and the coefficient of $$x^{n - 1}$$ is n( n + 3 )/2 , ( n = 1, 2 , . . . ) . Then S equals:

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