USEFUL FORMULAS. 



I. Algebraic. 



a. Arithmetic and geometric means. The arithmetic mean of n quanti- 

 ties a, i>, c, . . . is 



71 



{a^b^c^... ); 



their geometric mean is 



A case of special interest is 



{a b c. .)''. 



A^ab=.\{a-\-b) I I 



(i^:)i- 



b. Arithmetic progression. If a is the first term, and a-\-d, a -{- 2 d, 

 a -j- 3 ^/, ... are the successive terms, the ;?th or last term z is 



z =^ a -\- {ii — \) d. 



The sum s of the ?i terms of this series is 



s =^ \ {a -\- z) }i =^ {a -\- \ {ii — \) d') 11 

 ^=- {z — ^ (^^ — i) d^ n 



c. Geometric progression. If a is the first term, and a r, a r^, . . . are the 

 successive terms, the «th or last term z is 



a r 



n — \ 



The sum of the n terms is 



a if^— i) r z 



a 



If 



z (^'— i) 

 r—\ r—\ (r— I)r"~'• 



r < I and // = CO, 

 a 



\ — r 



d. Sums of special series. 



I + 2 + 3 + 4 + • • • + « —\n{iiA^x) 



2+4+6+8+ 



1+3+5+7+ 

 i^+ 2^+ f^ 4-+ 

 i'+ 2«+ 3'+ 4^+ 



+ 2 « = « {11 + i) 



-\- (2 n — i) = «^ 



-\-n- =^n{n+i) (2 n + i) 



+ «3 =1 ^^2 (« + if. 



