Formulas of Sequence and Series

    • The nth term {a_n} of the Arithmetic Progression (A.P) a,{\text{ }}a + d,{\text{ }}a + 2d, \ldots is given by {a_n} = a + (n - 1)d.

     

    • Arithmetic mean between a and b is given by A.M = \frac{{a + b}}{2}.

     

    • If {S_n} denotes the sum up to n terms of A.P. a,{\text{ }}a + d,{\text{ }}a + 2d, \ldots then {S_n} = \frac{n}{2}(a + l) where l stands for last term, {S_n} = \frac{n}{2}[2a + (n - 1)d]

     

    • The sum of n A.M’s between a and b is  = \frac{{n(a + b)}}{2}.

     

    • The nth term {a_n} of the geometric progression a,{\text{ }}ar,{\text{ }}a{r^2},{\text{ }}a{r^3},  \ldots is {a_n} = a{r^{n - 1}}.

     

    • Geometric mean between a and b is  G.M  = \pm \sqrt {ab} .

     

    • If {S_n} denotes the sum up to nterms of G.P is {S_n} = \frac{{a(1 - {r^n})}}{{1 - r}};{\text{ }}r \ne 1, {S_n} = \frac{{a - rl}}{{1 - r}};{\text{ }}l = a{r^n} where \left| r \right| < 1

     

    • The sum S of infinite geometric series is S = \frac{a}{{1 - r}};{\text{ }}\left| r \right| < 1

     

    • The nth term {a_n} of the harmonic progression is {a_n} = \frac{1}{{a + (n - 1)d}}.

     

    •  Harmonic mean between a and b is H.M = \frac{{2ab}}{{a + b}}.

     

    • {G^2} = A \cdot H and A > G > H; where A,G,H are usual notations.