# What is the sum of the arithmetic series 1,2,3,4...80?

##### 3 Answers

It is an arithmetic series with first term

hence

#### Explanation:

The sum of a finite arithmetic sequence is equal to the number of terms multiplied by the average term. The average term is the same as the average of the first and last term.

So in our example:

#sum_(n=1)^80 n = 80 * (1+80)/2 = 40*81 = 3240#

#### Explanation:

A good way to envision how to do this is to imagine pairs:

Start with the largest and smallest terms of the sequence:

#80+1=81#

The next largest and smallest are

#79+2=81#

We can start listing these pairs:

#80+1=81#

#79+2=81#

#78+3=81#

#77+4=81#

#76+5=81#

#..."continue"...#

#43+38=81#

#42+39=81#

#41+40=81#

In total, there are