Formula for Arithmetic Series
The Simple Arithmetic Sequences Let's say we have the simplest of arithmetic sequences. $\{1, 2, 3, \cdot\cdot\cdot, n\}$ And what I want to think about is what is the sum of this sequence going to be? And the sum of a sequence, we already know we call a series as following: $S_n = 1 + 2 + 3 + \cdot\cdot\cdot + n$ $S_n = n + (n-1) + (n-2) + \cdot\cdot\cdot + 1$ Now I'm going to add these two equations. $2S_n = (n+1) + (n+1) + (n+1) + \cdot\cdot\cdot + (n+1)$ So how many of these $(n+1)$ do we have? Well we have n of them there were n of these terms in each of these equations. So, we can rewrite this thing as following: $2S_n = n(n+1)$ $$ S_n = \frac{n(n+1)}{2} = n \cdot \frac{n+1}{2} = n \cdot \frac{a_n+a_1}{2} $$ $a_n$ is the nth term in our sequence, $a_1$ is the first term in our sequence. General Arithmetic Sequences Let's write an arithmetic sequence in general terms. $\{a, a+d, a+2d,\cdot\cdot\cdot, a+(n-1)d\}$ d could be a positive or a negative number, which we cal