The terms of an arithmetic sequence are generated by adding a fixed term [tex]r[/tex] every time.
So, we start with [tex]a_1=15[/tex], and we continue with [tex]a_2=15+r[/tex], [tex]a_3=15+2r[/tex] and so on.
As you can see, the general rule is [tex]a_n = 15+(n-1)r[/tex]
With this information, we can derive [tex]r[/tex], knowing that
[tex]a_{100} = 307 = 15+99r \iff 99r = 292 \iff r = \dfrac{292}{99}[/tex]
So, the sum of the first 100 terms is
[tex][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}[/tex]
