Sequence and Series

Arithmetic progression (AP)

nth term
$T_{n} = a + (n - 1) d,$ $T_{n} = S_{n} - S_{n - 1}$
Sum
$S_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} (a + l)$
Three terms
$a, b, c$ in AP $⟺ 2 b = a + c;$ take $a - d, a, a + d$
Arithmetic mean
$A = \frac{a + b}{2};$ $n$ AMs between $a, b : d = \frac{b - a}{n + 1}$

nth term
$T_{n} = a r^{n - 1}$
Sum
$S_{n} = \frac{a ( r ^{n} - 1 )}{r - 1} (r \neq = 1),$ $S_{\infty} = \frac{a}{1 - r} (∣ r ∣ < 1)$
Geometric mean
$a, b, c$ in GP $⟺ b^{2} = a c;$ $G = ab;$ $n$ GMs: $r = (\frac{b}{a})^{\frac{1}{n + 1}}$

HP
$a, b, c$ in HP $⟺ \frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ in AP $⟺ b = \frac{2 a c}{a + c}$
AM ≥ GM ≥ HM
$A \geq G \geq H (a, b > 0),$ $G^{2} = A H$ (two numbers)
AM–GM for n numbers
$\frac{a _{1} + a _{2} + \dots + a _{n}}{n} \geq n a_{1} a_{2} \dots a_{n},$ equality iff all equal

Natural numbers
$k = 1 \sum n k = \frac{n ( n + 1 )}{2},$ $\sum k^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6},$ $\sum k^{3} = [\frac{n ( n + 1 )}{2}]^{2}$
AGP sum to infinity
$a + (a + d) r + (a + 2 d) r^{2} + \dots = \frac{a}{1 - r} + \frac{d r}{( 1 - r ) ^{2}} (∣ r ∣ < 1)$
Method of differences
$T_{k} = f (k) - f (k - 1) \Rightarrow k = 1 \sum n T_{k} = f (n) - f (0)$
If the differences of successive terms form an AP or GP, write $T_{n}$ as a polynomial or exponential in $n$ first.