GATE CSE
First time here? Checkout the FAQ!
x
+1 vote
169 views

The solution of the recursive relation 
for n ≥ 2, a0 = 1, a1 = 0
image:EM4/Q10a.PNG

  1.  image:EM4/O10a.PNG
  2.   image:EM4/O10b.PNG

     
  3.   image:EM4/O10c.PNG
  4.   image:EM4/O10d.PNG
asked in Combinatory by Veteran (13k points)   | 169 views

2 Answers

+3 votes
Best answer
Put n = 1 we get $\left ( \frac{1}{2} \right )^{3} - \left ( \frac{-1}{2} \right )^{3} = \left ( \frac{1}{8} \right ) + \left ( \frac{1}{8} \right ) = \left ( \frac{1}{4} \right )$ which is matched with given reccurence relation .= 1/4

Put n= 2 we get $\left ( \frac{1}{2} \right )^{4} - \left ( \frac{-1}{2} \right )^{4} = \left ( \frac{1}{16} \right ) - \left ( \frac{1}{16} \right ) = \left ( 0 \right )$  which is matched with given reccurence relation .= 0

Put n= 3 we get $\left ( \frac{1}{2} \right )^{5} - \left ( \frac{-1}{2} \right )^{5} = \left ( \frac{1}{32} \right ) + \left ( \frac{1}{32} \right ) = \left ( \frac{1}{16} \right )$  which is matched with given reccurence relation .= (1/4 )/4 = (1/16)

So B is answer
answered by Veteran (47.4k points)  
selected by

by putting n=2 ,we will get 1/4 as a0 =1

and answer given is B actually
now check always try to put in these quetion :) but no wrongly like i did :)
thanks a lot @anirudh..
+1 vote

4 an - an-2 = 0 ...............from the given recurrance

Now, adding and subtracting an-3,

\therefore  4 an - an-2 + an-3 - an-3 = 0

\therefore  4 an - an-2 + an-3 4 an-1 = 0      (from the recurrance for an-1)

\therefore   4 an 4 an-1 - an-2 + an-3= 0  

This is  our normal recurrance now.

\therefore 4t3 - 4t2 - t +1 =0

By trial and error, 1/2 is one of the factors.

Now, we perform synthetic division.

1/2  |     4    -4       -1      1

        |          +2       -1      -1

---------------------------------------

            4     -2       -2       0

So, factorisation is

(t-1/2)(t -1)( 2t +1) = 0

So, factors are t= 1/2 , -1/2, 1

So, f(n) = c1 . (1/2)n + c2. (1)n + c3. (-1/2)n

 

Now, putting the values of a0 and a1,   we get answer B.

answered by Veteran (15k points)  


Top Users May 2017
  1. akash.dinkar12

    3292 Points

  2. pawan kumarln

    1652 Points

  3. sh!va

    1650 Points

  4. Arjun

    1424 Points

  5. Bikram

    1372 Points

  6. Devshree Dubey

    1272 Points

  7. Debashish Deka

    1142 Points

  8. Angkit

    1044 Points

  9. LeenSharma

    904 Points

  10. srestha

    718 Points

Monthly Topper: Rs. 500 gift card
Top Users 2017 May 22 - 28
  1. Bikram

    458 Points

  2. Arnab Bhadra

    402 Points

  3. pawan kumarln

    278 Points

  4. Ahwan

    236 Points

  5. bharti

    194 Points


22,786 questions
29,121 answers
65,184 comments
27,661 users