Finite series formulas.

1. Arithmetic series.
   
   $$a+ (a+d) + \cdots +(a+(n-1)d) = \sum_1^n (a+(i-1)d) = \frac{n(2a+(n-1)d)}{2}.$$
   
   

2. Geometric series formula.
   
   $$a+ ar + \cdots ar^{n-1} = \sum_1^n ar^{i-1} =a \frac{1-r^n}{1-r}.$$
   
   

3. Telescoping series formula. If $g(x)=f(x)-f(x-1)$, then

   
   

   $$\sum_1^n g(i) = f(n)-f(0).$$
   
   

4. Sums of powers.
   1. 
      
      $$\sum_1^n 1 = \frac{n(n+1)}{2}.$$
      
      

   2. 
      
      $$\sum_1^n i = \frac{n(n+1)}{2}.$$
      
      

   3. 
      
      $$\sum_1^n i^2 = \frac{n(n+1)(2n+1)}{6}.$$
      
      

   4. 
      
      $$\sum_1^n i^3 = \frac{n^2(n+1)^2}{4}.$$
      
      

   5. 
      
      $$\sum_1^n i^4 = \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}.$$