GATE CSE
First time here? Checkout the FAQ!
x
+1 vote
118 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 (11.1k points)   | 118 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 (44.5k 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 (14.7k points)  
Top Users Feb 2017
  1. Arjun

    5224 Points

  2. Bikram

    4230 Points

  3. Habibkhan

    3748 Points

  4. Aboveallplayer

    2986 Points

  5. Debashish Deka

    2356 Points

  6. sriv_shubham

    2298 Points

  7. Smriti012

    2142 Points

  8. Arnabi

    2008 Points

  9. sh!va

    1654 Points

  10. mcjoshi

    1628 Points

Monthly Topper: Rs. 500 gift card

20,832 questions
25,989 answers
59,623 comments
22,046 users