Express numbers, raised to an exponent, for sum of consecutive numbers.

[mente oscura]

Hello.

Not long ago, I did a study on numbers, raised to an exponent. I noticed that a "pattern" remained, and I could find a general formula.

Let: $$a , n, k \in{N}$$, when "a" is odd number:

I define: $$a _1, \ldots , a_k \in{N}$$, as the consecutive addends, such that:

Let: $$a , n, k \in{N}$$, when "a" is even number:

I define: $$a _1, \ldots , a_k \in{Q}$$, as the addends, such that: $$a_{i+1}-a_i=1$$.Two cases are presented to us:

1º) If “n”= even:

$$a_1= \dfrac{a^{\frac{n}{2}}+1}{2}$$

$$a_k= \dfrac{3a^{\frac{n}{2}}-1}{2}$$

$$k=a^{\frac{n}{2}}$$

Demonstration:

$$a^n=\dfrac{\dfrac{a^{\frac{n}{2}}+1}{2}+\dfrac{3a^{\frac{n}{2}}-1}{2}}{2}a^{\frac{n}{2}}=$$

$$=\dfrac{\dfrac{4a^{\frac{n}{2}}}{2}}{2}a^{\frac{n}{2}}= a^n$$2º) If “n” =odd:

$$a_1= \dfrac{a^{\frac{n+1}{2}}-2a^{\frac{n-1}{2}}+1}{2}$$

$$a_k= \dfrac{a^{\frac{n+1}{2}}+2a^{\frac{n-1}{2}}-1}{2}$$

$$k=2a^{\frac{n-1}{2}}$$

Demonstration:

$$a^n=\dfrac{(\dfrac{a^{\frac{n+1}{2}}-2a^{\frac{n-1}{2}}+1}{2})+{(\dfrac{a^{\frac{n+1}{2}}+2a^{\frac{n-1}{2}}-1}{2})}}{2}2a^{\frac{n-1}{2}}=$$

$$=\dfrac{2a^{\frac{n+1}{2}}}{4}2a^{\frac{n-1}{2}}=a^n$$

Examples:

1º) n=6, a=5

$$a_1=\dfrac{5^{\frac{6}{2}}+1}{2}=63$$

$$a_k=\dfrac{3*5^{\frac{6}{2}}-1}{2}=187$$

$$k=5^{\frac{6}{2}}=125$$

$$5^6=63+64+65+...+186+187=\dfrac{63+187}{2}125=15625$$

2º) n=7. a=4

$$a_1=\dfrac{4^{\frac{7+1}{2}}-2*4^{\frac{7-1}{2}}+1}{2}=64.5$$

$$a_k=\dfrac{4^{\frac{7+1}{2}}+2*4^{\frac{7-1}{2}}-1}{2}=191.5$$

$$k=2*4^{\frac{7-1}{2}}=128$$

$$4^7=64.5+65.5+66.5+...+190.5+191.5=\dfrac{64.5+191.5}{2}128=16384$$Note: I do not know if it will be "invented". If it is not, I could call it: "mente oscura" Theorem.(Rofl)

Regards.

 

[jvicens]

Hello there,

Thank you for sharing your study on numbers raised to an exponent. It is always exciting to see patterns emerge in mathematics and to find general formulas that can explain them.

I did take a look at your examples and the formulas you have presented seem to hold true. However, I would suggest further testing and analysis to ensure that they hold true for all values of a, n, and k.

Also, I would like to point out that the term "mente oscura" translates to "dark mind" in English. I am not sure if that is the intended name for your theorem, but it is always important to choose names that accurately represent the concept being studied.

Overall, great work on your study and I encourage you to continue exploring and discovering patterns in mathematics.