How do you find the sum of the series #n# from n=1 to 5?

1 Answer
Jul 5, 2018

Answer:

#S_n = 15 #

Explanation:

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