# Week 1 – precalculus 11

In my first week of  Pre-Calculus 11 I learned some interesting concepts. However, the concept of an Arithmetic Series intrigued me the most. An Arithmetic Series is the sum of all terms in a sequence in which the numbers are increasing or decreasing by the same amount in the next place in the sequence. This means that one could easily add up every number from 1 to 100 with just the 2 values. The reason this is possible is because 1+100=101 and that if one continues to add together the next number from 1 and the next number below 100, it will still equal 101 until you have 50 pairs that add up to 101. The equation for a problem like this would look like the following: 50(101) The 50 is the amount of pairs and the 101 is what each pair adds up to. This makes a problem that would take a long time relatively little time to complete. This can also be used for a sequence that increases or decreases by the same amount each consecutive number. All we need to complete an equation like this is the first term(number) in the sequence, $t_1$, and the last term in the sequence we will add until. In this case we will use the 10th term in the sequence,$t_{10}$. The equation to find the sum of the first 10 terms in the sequence, represented by $S_{10}$, is $S_{n} = \frac{n}{2}(t_1 + t_n)$. n represents the amount of terms being added or subtracted together. The reason we multiply the added pair by n is to include all the terms in the equation and we divide this by 2 to make them pairs. In this example we will make the first term 7 and the 10th term 25, increasing each number by 2 in the sequence. Our equation would look like this:

$S_{n} = \frac{n}{2}(t_1 + t_n)$

Substitute: n=10 $t_1 = 7$  $t_{10} = 25$

$S_{10} = \frac{10}{2}(7 + 25)$

$S_{10} = \frac{10}{2}(32)$

$S_{10} = \frac{320}{2}$

$S_{10} = 160$

The first 10 terms in this sequence added together equals 160