Solving recurrence relation using Mathematica

Question

Good evening, experts

I want to solve recurrence equation using mathematica,

x(n) = x(n − 1) + n 
for n > 0, 
    x(0) = 0

And i need to find x(1), x(2), x,(3)

This is my input and it gives me errors

n > 0
a[0] := 0
RSolve[x == a[n - 1] + n, a[n], n]

How can I rewrite the equation using the mathematica? Thanks in advance

Answer 1

An example of this very pattern is the 2nd example in the documentation for RSolve:

Include a boundary condition:

In[1]:= RSolve[{a[n + 1] - 2 a[n] == 1, a[0] == 1}, a[n], n]

Out[1]= {{a[n] -> -1 + 2^(1 + n)}}

For your problem, that'd be:

In[1]:= RSolve[{a[n] == a[n - 1] + n, a[0] == 0}, a[n], n]

Out[1]= {{a[n] -> 1/2 n (1 + n)}}    

Answer 2

Simply use

RSolve[{a[n] == a[n - 1] + n, a[0] == 0}, a[n], n]

Remove the following:

n > 0
a[0] := 0

a[0] := 0 is a function definition. a must not have associated definitions in order to work in RSolve

Answer 3

If you want to find x(1), x(2), x(3), you can use RecurrenceTable:

RecurrenceTable[{x[n] == x[n - 1] + n, x[0] == 0}, x[n], {n, 3}]

{0,1,3,6}

x(1)=1, x(2)=3, x(3)=6