# Interesting Relations Between Numbers

Hi Everyone .

I had been working out on numbers for quite some time and I had got some interesting relations among numbers which I will write below .

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Relation 1 :

For any three consecutive integers $$a, b, c$$

$$\color{blue}\boxed{b^2 = (a \times c) + 1}$$

Example : let us take the numbers 24, 25, 26

$$25^2 = 24 \times 26 + 1 \Rightarrow 625 = 624 + 1 \Rightarrow 625 = 625$$

Relation 2 :

For any three consecutive integers $$a, b, c$$

$$\color{red}\boxed{(a \times b) + 2c + a = c^2}$$

Example : let us take 3 numbers 3, 4, 5

$$(3 \times 4) + 2 \times 5 + 3 = 5^2 \Rightarrow 12 + 10 + 3 = 25 \Rightarrow 25 = 25$$

Relation 3 :

For any two consecutive integers $$a, b$$

$$\color{yellow}\boxed{(a \times b) - a = a^2}$$

Example : let us take two numbers 6, 7

$$(6 \times 7) - 6 = 42 - 6 = 36 = 6^2$$

Relation 4 :

For any three consecutive numbers $$a, b, c$$

$$\color{orange}\boxed{a^2 - b^2 + 2c = 3}$$ (OR) $$\color{orange}\boxed{(a + b)(a - b) + 2c = 3}$$

Example : let us take three numbers 7, 8, 9

$$(7 + 8)(7 - 8) + 2(9) = -15 + 18 = 3$$

Relation 5 :

For any two consecutive even or odd numbers $$a, b$$

$$\color{green}\boxed{a^2 + 2a + 2b = b^2}$$ (OR) $$\color{green}\boxed{a^2 + 4a + 4 = b^2}$$

Example : let us take two consecutive odd numbers 13, 15

$$13^2 + 2(13) + 2(15) = 169 + 26 + 30 = 225$$

Relation 6 :

For any two consecutive numbers $$a, b$$ such that $$a > b$$

$$\color{brown}\boxed{a^2 - b^2 = a + b}$$ (OR) $$\color{brown}\boxed{b^2 = a^2 - (a + b)}$$ (OR) $$\color{violet}\boxed{a^2 = b^2 + a + b}$$

Example : let us take two numbers 31, 30

$$31^2 - 30^2 = 961 - 900 = 61 = 31 + 30$$

Relation 7 :

If there are three whole numbers $$a, b, c$$ such that $$\color{blue}a \times b = c$$ , then

\begin{align} a^2 \times b^2 = c^2\\ a^3 \times b^3 = c^3\\ a^4 \times b^4 = c^4\\ a^5 \times b^5 = c^5\\ \end{align}

So , if $$\color{violet}a \times b = c$$ then $$\color{green}\boxed{a^n \times b^n = c^n}$$

Example : let us take the numbers 2, 3, 6 because $$2 \times 3 = 6$$

So , now \begin{align} 2^2 \times 3^2 = 6^2 = 36\\ 2^3 \times 3^3 = 6^3 = 216\\ 2^4 \times 3^4 = 6^4 = 1296\\ 2^5 \times 3^5 = 6^5 = 7776\\ \end{align}

Relation 8 :

Sum of any $$n$$ consecutive integers is given by :

$$\color{red}\boxed{Sum = Median \times n}$$

Example : find the sum of 11, 12, 13, 14, 15

n = 5 (odd)

Median = $$\frac{5 + 1}{2} = \frac{6}{2} = 3^{th}$$ term = 13

$$Sum = median \times n = 13 \times 5 = 65$$

That's all for now . Hope these relations will help you . If any mistakes are there please inform me .

If you too have any kind of relations with you please put it in form of a comment below and I will write them in this note in your name .

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Note by Ram Mohith
5 days, 13 hours ago

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