# Mathematics

Contents

## Miscellaneous

Phi ($$\phi$$) is the golden ratio.

$${\Large \phi = \frac{ 1 + \sqrt{5} }{2}}$$

Look into Toads and Frogs game. See OEIS “A005563 a(n) = n*(n+2) = (n+1)^2 – 1″ or OEIS search for “frog”.

x^2+(5y/4-sqrt(abs(x)))^2=1

x^2+(5y/4-sqrt(abs(x)))^2=1

## Numbers

$$3^3 + 4^4 + 3^3 + 5^5 = 3435$$ — The only other number with this property is 1.

## Notation

### $${}^n C {}_r$$ — N Choose R

$${}^n C {}_r$$ (“n choose r”) is an alternate notation for the binomial symbol:
$${}^n C {}_r = {n \choose r} = \frac{n!}{r!(n−r)!}$$

$${}^n C _r$$ is an alternate notation for the binomial symbol $${n \choose r} = \frac{n!}{r!(n−r)!}$$

Inline style: $${}^{10}C_{3} = {10 \choose 3} = \frac{10!}{3!(10−3)!} = \frac{10!}{3! 7!} = \frac{3628800}{6×5040} = \frac{3628800}{30240} = 120$$
$${}^{10}C_{3} = {10 \choose 3} = \frac{10!}{3!(10−3)!} = \frac{10!}{3! 7!} = \frac{3628800}{6×5040} = \frac{3628800}{30240} = 120$$

Display style: $${}^{10}C_{3} = {10 \choose 3} = \frac{10!}{3!(10−3)!} = \frac{10!}{3! 7!} = \frac{3628800}{6×5040} = \frac{3628800}{30240} = 120$$
$${}^{10}C_{3} = {10 \choose 3} = \frac{10!}{3!(10−3)!} = \frac{10!}{3! 7!} = \frac{3628800}{6×5040} = \frac{3628800}{30240} = 120$$

### $$\sum_{n=1}^{\infty}$$ — Summation

Inline style: $$\sum_{n=1}^{\infty} 2^{-n} = 1$$
$$\sum_{n=1}^{\infty} 2^{-n} = 1$$

Display style: $$\sum_{n=1}^{\infty} 2^{-n} = 1$$
$$\sum_{n=1}^{\infty} 2^{-n} = 1$$

Inline style: $$\sum\limits_{k=1}^{n} k$$
$$\sum\limits_{k=1}^{n} k$$

Display style: $$\sum\limits_{k=1}^{n} k$$
$$\sum\limits_{k=1}^{n} k$$

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