# APM - Q2 - Finding a general result

> Source: https://ollybritton.com/notes/maths/problems/advanced-problems-in-mathematics/q1/ · Updated: 2021-10-25 · Tags: advanced-problems-in-mathematics, notes, maths

## Flashcards
### 2021-10-25
##### $$\frac{1 + 4}{2} = \frac{1 + 2 + 3 + 4}{4}$$ What general rule is this??
That the average of two numbers is also the average of the sum between them.

##### $$\frac{b^2 + b - a^2 + a}{b - a + 1}$$ What is the trick to simplifying this??
Realising that it's factorising but one bracket has three terms.

$$
\frac{(b - a + 1)(a + b)}{b -a + 1}
$$

##### What's a trick for coming up with a formula for the sum of $k\text{th}$ powers over the integers??
Assuming that it will be equal a $(k + 1)$ degree polynomial and coming up with 5 equations.

---
Olly Britton — https://ollybritton.com. Machine-readable index: https://ollybritton.com/llms.txt