The formula for the sum of the series $1+2+\cdots +n$ is $n(n+1)/2$. What if we weren’t sure that was the right function? How could we check? Using what we learned about functions we can create one that computes the $S_n$:

                
            compute_s_n <- function(n){
  x <- 1:n
  sum(x)
}

How can we compute $S_n$ for various values of $n$, say $n=1,\dots ,25$? Do we write 25 lines of code calling compute_s_n? No, that is what for-loops are for in programming. In this case, we are performing exactly the same task over and over, and the only thing that is changing is the value of $n$. For-loops let us define the range that our variable takes (in our example $n=1,\dots ,10$), then change the value and evaluate expression as you loop.

Perhaps the simplest example of a for-loop is this useless piece of code:

                
            for(i in 1:5){
  print(i)
}
#> [1] 1
#> [1] 2
#> [1] 3
#> [1] 4
#> [1] 5

Here is the for-loop we would write for our $S_n$ example:

                
            m <- 25
s_n <- vector(length = m) # create an empty vector
for(n in 1:m){
  s_n[n] <- compute_s_n(n)
}

In each iteration $n=1,2,\dots$ we compute $S_n$ and store it in the $n^{th}$ entry of s_n.

Now we can create a plot to search for a pattern:

                
            n <- 1:m
plot(n, s_n)

If you noticed that it appears to be a quadratic, you are on the right track because the formula is $n(n+1)/2$. which we can confirm with a table:

                
            head(data.frame(s_n = s_n, formula = n*(n+1)/2))
#>   s_n formula
#> 1   1       1
#> 2   3       3
#> 3   6       6
#> 4  10      10
#> 5  15      15
#> 6  21      21

We can also overlay the two results by using the function lines to draw a line over the previously plotted points:

                
            plot(n, s_n)
lines(n, n*(n+1)/2)