Sequence & Series

Formula Sheet 6

1. Arithmetic Sequence (A.P.)

Parameter Formula
General Term ($$ a_n $$) $$ a_n = a + (n-1)d $$
Sum of n terms ($$ S_n $$) $$ S_n = \frac{n}{2}[2a + (n-1)d] $$
Sum (Last Term Known) $$ S_n = \frac{n}{2}(a + l) $$
Arithmetic Mean (A.M.)
(of a and c)		 $$ b = \frac{a+c}{2} $$

2. Geometric Sequence (G.P.)

Parameter Formula
General Term ($$ a_n $$) $$ a_n = a \cdot r^{n-1} $$
Sum ($$ S_n $$) if $$ r > 1 $$ $$ S_n = \frac{a(r^n – 1)}{r-1} $$
Sum ($$ S_n $$) if $$ r < 1 $$ $$ S_n = \frac{a(1 – r^n)}{1-r} $$
Infinite Sum ($$ S_{\infty} $$)
(if $$ |r| < 1 $$)		 $$ S_{\infty} = \frac{a}{1-r} $$
Geometric Mean (G.M.) $$ b = \sqrt{ac} $$

3. Special Series (Sum of n terms)

First n numbers ($$ \sum n $$)

$$ \frac{n(n+1)}{2} $$

Sum of Squares ($$ \sum n^2 $$)

$$ \frac{n(n+1)(2n+1)}{6} $$

Sum of Cubes ($$ \sum n^3 $$)

$$ \left[\frac{n(n+1)}{2}\right]^2 $$
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