GATE | Sudo GATE 2020 Mock I (27 December 2019) | Question 20

Consider the recurrence relation a_n – 11a_n-1 + 30a_n-2 = 0 n>1 , a₀ = 1, a₁ = 8. The value of a₄ is ______________ .
(A) 2638
(B) 2683
(C) 3826
(D) 8362


Answer: (A)

Explanation: Characteristics equation,

C(t)= x2 - 11x +30 = 0 
(x-5)(x-6)=0
x=5,6 

General solution,

an = C1 5n + C2 6n
a0 = C1 50 + C2 60 =1
C1  + C2 =1  ... ... (1)
a1 =  Cn 51 + Cn 61 =8
5C1  + 6C2= 8 ... ... (2) 

From (1) and (2),

5C1  + 6C2= 8
5C1  + 5C2 =5
C2=3

And C1 = -2 

Therefore,

an = -2*5n + 3*6n 

The value of a₄ is,

= -2*54 + 3*64
= -1250 + 3888
= 2638 

Option (A) is correct.

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