Question

Find the sum of the first 100

Answer 1

The terms of an arithmetic sequence are generated by adding a fixed term $r$ every time.

So, we start with $a_1=15$, and we continue with $a_2=15+r$, $a_3=15+2r$ and so on.

As you can see, the general rule is $a_n = 15+(n-1)r$

With this information, we can derive $r$, knowing that

$a_{100} = 307 = 15+99r \iff 99r = 292 \iff r = \dfrac{292}{99}$

So, the sum of the first 100 terms is

$[tex]\displaystyle \sum_{i=0}^{99} 15+i\dfrac{292}{99} = \displaystyle \sum_{i=0}^{99} 15 + \displaystyle \dfrac{292}{99}\sum_{i=0}^{99} i = (15\cdot 99) + \dfrac{292}{99}\dfrac{99\cdot 100}{2} = 1485 + \dfrac{490342}{99}$