GATE Overflow Test Series | Discrete Mathematics | Test 2 | Question: 28

Question

The $n^{th}$ term of the sequence generated by the function $\frac{2}{(1-x)^2} \cdot \frac{x}{1-x-x^2}$ is represented as $a_n$. The value of $a_6$ is$\_\_\_\_\_\_$.

Comments

if we are talking about a6 we mean that it’s 6^th term right?
so why aren’t we finding the coefficient of x^5 here as it’s the 6^th term in the series coz first term is x^0 right? correct me if I am wrong plss.

Answer 1

The first sequence is given by $2(1,2,3,4,5,\ldots) = 2,4,6,8,10,\ldots$

The second sequence is the Fibonacci sequence given by $0,1,1,2,3,5,8,\ldots$

Multiplying the generating functions of both the sequences we get,

$(2 + 4x + 6x^2 + 8x^3 + 10x^4+12x^5+14x^6 \ldots) (0+x+x^2+2x^3+ 3x^4 + 5^5+8x^6\ldots)$

Here, we have to find the coefficient of $x^6$ which is given by

$2\times 8 + 4 \times 5 + 6 \times 3 + 8 \times 2 + 10 \times 1 + 12 \times 1  $

$ = 16 + 20 + 18 + 16 + 10 + 12 = 92$

Comments

how first sequence came?

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Is there any way to identify sequence ?

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