Assuming that you mean that each incremental increase is 1, this arithmetic series can be calculated. In an arithmetic series, there are a few fundamental variables that must be determined in order to solve the problem (we'll look at these later).
Firstly, let's start with the formula for an arithmetic series:
S_n = (n/2)*(a + t_n)
Alternatively,
S_n = n/2 [2a + (n-1)d]
Upon simple analysis, you will notice that the second formula uses a new variable, d. We know that since these two formulas are equivalent, the variable d must belong to the variable present in formula 1, but not present in formula 2, t_n.
The term t_n is actually another, rather similar, type of formula. It is known as the arithmetic sequence formula.
The formula used to determine an arithmetic sequence (the arithmetic sequence formula), is given as:
t_n = a + (n-1) * d
For simplicity, we'll use the first formula. We know that in the sequence:
1, 2, 3 ... 5
The first term is 1, and the last term is 5. In this case, where in use of the term t_n, we define n (within t_n), as the number of terms in the series. So, in this case t_n = 5.
Remember we said that we'd define those variables? Let's do that now. Firstly (in formula 1), we define a as the first term in the series. Likewise, n, is the last term in the series. Finally, of course, we know t_n. Within t_n, there is one term not defined, d. d is simply the common difference, or the constant difference between each term in the series.
So, we can now calculate the formula. We'll start with t_n
t_n = 5 = 1 + (n-1) * 1
Now, simply solve for n
5 - 1 = (n-1) * 1
= 4 / 1 = (n-1)
= 4 = n -1
= 5 = n
So, we've verified n.
Now, simply plug n into formula 1.
S_5 = 5/2 * (1 + 5) = 15
Put simply, in this arithmetic series, the sum of the first 5 terms is 15.
All the best,
Eden
