Answer:
(a)x(n)=5(n-1)
(b)x(n)=4 X 3ⁿ⁻¹
(c)x(n)=n(n+1)/2
Step-by-step explanation:
(a)x(n) = x(n − 1) + 5 for n > 1, x(1) = 0
Put n=n-1
x(n-1)=x(n-2)+5
Recall: x(n-1)= x(n)-5
Substitute:
x(n)-5=x(n-2)+5
x(n)=x(n-2)+5+5
Put n=n-2
x(n-2)=x(n-3)+5
Recall: x(n-2)= x(n)-5-5
Substitute:
x(n)-5-5=x(n-3)+5
x(n)=x(n-3)+5+5+5
Generalising
x(n)=x(n-i)+5i for i<n
Put i=n-1
x(n)=x(n-(n-1))+5(n-1)
x(n)=x(1)+5(n-1)=0+5(n-1)
x(n)=5(n-1)
(b)x(n) = 3x(n − 1) for n > 1, x(1) = 4
Put n=n-1
x(n-1)=3x(n-2)
Recall: x(n-1)= x(n)/3
Substitute:
x(n)/3=3x(n-2)
x(n)=3 X 3x(n-2)
Put n=n-2
x(n-2)=3x(n-3)
Recall: x(n-2)= x(n)/3²
Substitute:
x(n)/3²=3x(n-3)
x(n)= 3³x(n-3)
Generalising
x(n)=3ⁱ x(n-i) for i<n
Put i=n-1
x(n)=3ⁿ⁻¹ x(n-(n-1))
x(n)=3ⁿ⁻¹ x(1)
x(n)=3ⁿ⁻¹*4
x(n)=4 X 3ⁿ⁻¹
(c) x(n) = x(n − 1) + n for n > 0, x(0) =0
Put n=1
x(1)=x(1-1)+1
x(1)=0+1=1
Put n=2
x(2)=x(2-1)+2
x(2)=x(1)+2=1+2=3
Put n=3
x(3)=x(3-1)+3=x(2)+3=3+3=6
Generalising
x(n)=n(n+1)/2
