What will be the sequence generated by the generating function 4x/(1-x)2?a)12, 16, 20, 24,…b)1, 3, 5, 7, 9,…c)0, 4, 8, 12, 16, 20,…d)0, 1, 1, 3, 5, 8, 13,…Correct answer is option 'C'. Can you explain this answer?

Question

What will be the sequence generated by the generating function 4x/(1-x)2?a)12, 16, 20, 24,&hellip;b)1, 3, 5, 7, 9,&hellip;c)0, 4, 8, 12, 16, 20,&hellip;d)0, 1, 1, 3, 5, 8, 13,&hellip;Correct answer is option 'C'. Can you explain this answer?

Answer

To find the sequence generated by the generating function, we can expand the function as a power series.

We have:

4x/(1-x)² = 4x * (1 + x + x² + x³ + ...)^2.

To find the coefficient of x^n in the expansion, we need to find the product of the coefficients of x^k in each term of the product, where k ranges from 0 to n.

For the term 4x * (1 + x + x² + x³ + ...)^2, the coefficient of x^k is the sum of the products of the coefficients of x^i and x^j, where i + j = k.

So, for the term 4x * (1 + x + x² + x³ + ...)^2, the coefficient of x^k is:

4 * (1 + 1 + 2 + 3 + ... + (k-1) + k) = 4 * (k(k+1)/2) = 2k(k+1).

Therefore, the sequence generated by the generating function 4x/(1-x)² is given by the coefficients of x^n, which are 2n(n+1) for n ≥ 0.

So, the sequence is:

2(0)(0+1), 2(1)(1+1), 2(2)(2+1), 2(3)(3+1), ...

which simplifies to:

0, 4, 12, 24, ...

Therefore, the sequence generated by the generating function 4x/(1-x)² is:

0, 4, 12, 24, ...

Answer

The sequence should be 0, 4, 8, 12, 16, 20,…for the generating function 4x/(1-x)², when basic generating function: 1/(1-x).

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