Sequences & Progressions
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Arithmetic Progressions
nth term
a_n = a + (n-1)d
$a$ is the first term, $d$ the common difference.
Sum of n terms
S_n = \frac{n}{2}\left[2a + (n-1)d\right]
Works for any AP.
Sum from first and last
S_n = \frac{n}{2}(a + l)
$l$ is the last term.
Middle term
\text{middle} = \frac{S_n}{n}
Only when the count $n$ is odd.
Gap when inserting k means
d = \frac{b - a}{k + 1}
$k$ means create $k + 1$ gaps.
Geometric Progressions
nth term
a_n = a r^{n-1}
$r$ is the common ratio; power is $n-1$.
Sum of n terms
S_n = \frac{a(r^n - 1)}{r - 1}
For $r \ne 1$; if $r = 1$, the sum is $na$.
Sum to infinity
S_\infty = \frac{a}{1 - r}
Only when $|r| < 1$.
Three GP terms
\frac{a}{r},\ a,\ ar
Their product is $a^3$.
Bouncing ball total
h + \frac{2hr}{1 - r}
$h$ is the drop height, $r$ the rebound fraction.
Harmonic Progressions & AGP
nth term of an HP
\frac{1}{a + (n-1)d}
$a$, $d$ come from the AP of reciprocals.
HM of two numbers
\frac{2ab}{a + b}
The reciprocal of the AM of the reciprocals.
AM x HM = GM squared
AM \times HM = GM^2
For positive numbers, AM $\ge$ GM $\ge$ HM.
Infinite AGP sum
\frac{a}{1-r} + \frac{dr}{(1-r)^2}
For $|r| < 1$, series $a + (a+d)r + (a+2d)r^2 + \cdots$
Special Series & Standard Sums
Sum of first n numbers
\sum_{k=1}^{n} k = \frac{n(n+1)}{2}
Sum of first n squares
\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}
Sum of first n cubes
\sum_{k=1}^{n} k^3 = \left[\frac{n(n+1)}{2}\right]^2
The square of the natural-number sum.
First n odd numbers
1 + 3 + \cdots + (2n-1) = n^2
The run must start at 1.
First n even numbers
2 + 4 + \cdots + 2n = n(n+1)
Telescoping split
\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}
Consecutive products
\sum_{k=1}^{n} k(k+1) = \frac{n(n+1)(n+2)}{3}
Pattern Sequences: Next, Missing & Wrong Terms
Common position rules
n^2,\ n^3,\ n^2+1,\ n^2-1,\ 2^n,\ n(n+1)
Compare each term with its position $n$.
Multiply-and-add rule
a_{n+1} = a_n \times r + c
A frequent hidden rule; test it when differences fail.