The solution to the recurrence relation is [tex]x(n) = \frac 12(3n - 1)[/tex]
The relation is given as:
x(n) = x(n/3) + n for n > 1
x(1) = 1
Calculate x(3)
x(3) = x(3/3) + 3
This gives
x(3) = x(1) + 3
x(3) = 1 + 3 = 4
Calculate x(9)
x(9) = x(9/3) + 9
This gives
x(9) = x(3) + 9
x(9) = 4 + 9 = 13
So, we have:
x(1) = 1
x(3) = 4
x(9) = 13
The above sequence forms an arithmetic sequence.
An arithmetic sequence is represented as:
x(n) = x(1) + (n - 1) * d
Substitute 9 for n
x(9) = x(1) + (9 - 1) * d
Substitute 13 for x(9) and 1 for x(1)
13 = 1 + (9 - 1) * d
This gives
12 = 8d
Divide by 8
d = 1.5
Substitute d = 1.5 in x(n) = x(1) + (n - 1) * d
x(n) = x(1) + (n - 1) * 1.5
Substitute x(1) = 1
x(n) = 1 + (n - 1) * 1.5
Expand
x(n) = 1 + 1.5n - 1.5
Evaluate the like terms
x(n) = 1.5n - 0.5
Express as fraction
x(n) = 3n/2 - 1/2
Factor out 1/2
[tex]x(n) = \frac 12(3n - 1)[/tex]
Hence, the solution to the recurrence relation is [tex]x(n) = \frac 12(3n - 1)[/tex]
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