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Series & Sequences
A-Level H2 Math
Arithmetic Progression (AP)
$T_n = a + (n-1)d$
$S_n = \dfrac{n}{2}(2a + (n-1)d) = \dfrac{n}{2}(a+l)$
Show sequence is AP: $T_{n+1} - T_n = d$ (constant)
Geometric Progression (GP)
$T_n = ar^{n-1}$
$S_n = \dfrac{a(1-r^n)}{1-r} = \dfrac{a(r^n - 1)}{r-1}$
$S_\infty = \dfrac{a}{1-r} \quad (|r| < 1)$
Show sequence is GP: $\dfrac{T_{n+1}}{T_n} = r$ (constant)
Finding the nth Term from Sum
$T_n = S_n - S_{n-1}$
Valid for $n \geq 2$. For $n = 1$: $T_1 = S_1$
Standard Summation Results
Sum of constant
$\sum_{r=1}^{n} a = an$
Sum of first n natural numbers
$\sum_{r=1}^{n} r = \dfrac{n}{2}(n+1)$
Sum of squares
$\sum_{r=1}^{n} r^2 = \dfrac{n}{6}(n+1)(2n+1)$
Sum of cubes
$\sum_{r=1}^{n} r^3 = \left[\dfrac{n}{2}(n+1)\right]^2$
GP sum example
$\sum_{r=1}^{n} 3^r = \dfrac{3(3^n-1)}{3-1} = \dfrac{3}{2}(3^n - 1)$
Replacing Dummy Variable / Replacing n
TechniqueOriginalAfter Substitution
Replace dummy variable
$r \to r+1$		$\displaystyle\sum_{r=1}^{n}\dfrac{1}{r(r+1)} = 1 - \dfrac{1}{n+1}$$\displaystyle\sum_{r=0}^{n-1}\dfrac{1}{(r+1)(r+2)} = 1 - \dfrac{1}{n+1}$
Replace $n$
$n \to n+1$		$\displaystyle\sum_{r=1}^{n} u_r = \dfrac{n}{n+1}$$\displaystyle\sum_{r=1}^{n+1} u_r = \dfrac{n+1}{n+2}$

Formulas for the Singapore A-Level H2 Mathematics syllabus