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

Jul 5, 2018

${S}_{n} = 15$

#### 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).

${S}_{n} = \left(\frac{n}{2}\right) \cdot \left(a + {t}_{n}\right)$

Alternatively,

${S}_{n} = \frac{n}{2} \left[2 a + \left(n - 1\right) d\right]$

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:

${t}_{n} = a + \left(n - 1\right) \cdot d$

For simplicity, we'll use the first formula. We know that in the sequence:

$1 , 2 , 3 \ldots 5$

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 ${t}_{n}$

${t}_{n} = 5 = 1 + \left(n - 1\right) \cdot 1$

Now, simply solve for $n$

$5 - 1 = \left(n - 1\right) \cdot 1$
$= \frac{4}{1} = \left(n - 1\right)$
$= 4 = n - 1$
$= 5 = n$

So, we've verified $n$.

Now, simply plug $n$ into formula 1.

${S}_{5} = \frac{5}{2} \cdot \left(1 + 5\right) = 15$

Put simply, in this arithmetic series, the sum of the first $5$ terms is $15$.

All the best,

Eden