What is the solution to the unsolved equation?

  • Thread starter kazimo
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In summary, the conversation is about finding the values of n and k in an equation where all the numbers from 1 to n are raised to the power of k and added, resulting in (n+1)^k. The participants discuss different solutions and possibilities, including a typo in the original equation and the possibility of solving it with negative numbers. They also mention a similar series and a formula for finding the sum of a geometric series.
  • #1
kazimo
2
0
Hey,
I am kinda new here but here's a problem for you guys:

There is an equation of the form:
((1)^k)+((2)^k)+...+((n-1)^k)+((n)^k) = ((n+1)^k)

This equation is such that all the numbers starting from1 till n are raised to the power of k and added and the result is (n+1)^k. What should n and k be?

Apart from ((1)^2) + ((2)^2) = ((3^2)) There isn't any other obvious answer. ( These were the first numbers I tried when I began trying to solve this problem.

Kazim
 
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  • #2
((1)^2) + ((2)^2) = ((3^2))?

Is this true? :confused:
 
  • #3
Clearly, it is not.

But 1+2=3 is true. And there's a more trivial solution : 1^0 = 2^0.
 
  • #4
rajesh said:
((1)^2) + ((2)^2) = ((3^2))?

Is this true? :confused:

It was a typo ...it should have been

(1^1)+(2^1)=(3^1)
 
  • #5
I just did a mistake.. it was supposed to be

((1)^1) + ((2)^1) = ((3)^1)

I will be careful in the future
 
  • #6
I believe that if k was an odd number and n was negative then it would be possible to solve it another way. With a negative number in there you can counteract all of the adding of things.
 
  • #7
is the original problem like this:

k
sigma (n^c) where c is any real constant
n=1

i was trying to figure that out, but if yours is

k
sigma (c^n) where c is any real constant,
n=1

then i think the sum is {[c^(k+1)]/(c-1)}-[1/(c-1)]

hope this helps
 
  • #8
I think about the closest series to that is 1+2+4+8++2^N =2^(n+1)-1. This comes about because [tex] 1 +r +r^2+r^n=\frac{-1+r^_(n+1)}{r-1}[/tex]
but, of course, 2-1=1, so the denominator disappears.
 
Last edited:

1. What is an "Unsolved Equation"?

An unsolved equation is a mathematical problem that has not yet been solved or proven to have a solution. It could be a complex equation that has not been solved by mathematicians, or a simpler equation that has not been solved for a specific variable.

2. Why are some equations considered unsolved?

Equations can be considered unsolved for a variety of reasons. Some equations may be too complex for current mathematical techniques to solve, while others may require advanced technology or computing power to find a solution. In some cases, an equation may also be unsolved because it has not yet been approached by mathematicians.

3. Can unsolved equations ever be solved?

Yes, it is possible for unsolved equations to be solved. Over time, advancements in mathematics and technology may lead to the solution of previously unsolved equations. In some cases, a new approach or perspective may also help to solve an unsolved equation.

4. How do unsolved equations impact the field of mathematics?

Unsolved equations are a driving force in the field of mathematics. They challenge mathematicians to think critically and develop new techniques and theories. The pursuit of solving these equations also leads to new discoveries and advancements in mathematics as a whole.

5. Are there any famous unsolved equations?

Yes, there are several famous unsolved equations, such as Fermat's Last Theorem, the Riemann Hypothesis, and the Navier-Stokes Equations. These equations have captivated mathematicians for centuries and continue to be actively researched and studied today.

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