GATE Overflow Test Series | Discrete Mathematics | Test 2 | Question: 24

Question

The generating function corresponding to the sequence with closed formula for its $n^{th}$ term as $a_n=4(6^n)+7(-2)^n$ is $\_\_\_\_\_$

• A. $\frac{4}{1-6x} + \frac{7}{1+2x}$
• B. $\frac{4}{1+6x} + \frac{7}{1-2x}$
• C. $\frac{6}{1-4x} + \frac{2}{1+7x}$
• D. $\frac{6}{1-4x} + \frac{2}{1-7x}$

Comments

This playlist covers everything you need to know to solve these kind of questions: link

Answer 1

The generating  function corresponding to the sequence with $n^{th}$ term given by $a^n$ is $\frac{1}{1-ax}$

So, for $4\times (6^n)$ we get  $\frac{4}{1-6x}$ and for $7\times (-2^n)$ we get $\frac{7}{1+2x}$

Adding both we get the generating function corresponding to  the sequence with closed formula for its $n^{th}$ term as $a_n=4(6^n)+7(-2)^n$ as

$\frac{4}{1-6x} + \frac{7}{1+2x}$

Ref: http://discrete.openmathbooks.org/dmoi2/section-27.html

Comments

can you give any relevant text for that so one can strengthen their concepts

---

@Rajat http://www.math.cmu.edu/~lohp/docs/math/2011-228/mit-ocw-generating-func.pdf