Question

1.    a) Recursively define a₀ = 1, a₁ = 3, a₂ = 5 and a_n = 3a_n-2 + 2a_n-3 for n ³ 3. Calculate a_n for n = 3,4,5,6

(4 Marks)

 

b) Find f(2), f(3), f(4) and f(5) for the following recursive functions.

                 f(0) = 1

f(1) = 2

f(k) = (f(k -1))² - f(k -2) + k²

(4 Marks)

 

Answer 1

$a_n=3a_{n-2}+2a_{n-3}$

n=3

$a_3=3x3+2x1=11$

n=4

$a_4=3x5+2x3=20$

n=5

$a_5=3x11+2x5=43$

n=6

$a_6=3x20+2x11=82$

b.$f(k)=(f(k-1))^2-f(k-2)+k^2$

$f(2)=4-1+4=7$

$f(3)=49-2+9=56$

$f(4)=3136-7+16=3145$

$f(5)=9891025-56+25=9890994$