f(n)=knf(n) = k^n for some nn?

To be specific, there have been questions on this topic:

n = 0 to 5 (inclusive)

n = 0 to 6 (inclusive)

This wiki contains spoliers about these questions, so you are recommended to do them first before reading the rest of this note.

In these questions: Actually, please do those questions first.

In these questions, k=2k=2 and n=0n = 0 to n=5n=5 and n=0n=0 to n=6n=6.

There's a lot of similarities, and even the answer is amazing, but I shall not spoil it.

Both questions are in the form "f(n)f(n) is a pp-degree polynomial. When q=0,1,2,,pq = 0,1,2,\ldots,p, f(q)=kq f(q) = k^q. What is f(x)f(x)?"

These questions are a special case of this form, with k=2k = 2 and x = 2*q+1. The answer is also very similar, f(x) = 2^(2*q).

I would like to know if this can be generalised for q and also, k, and if so, what values of x can create these values.

I'll work on it when I have time. Which doesn't come a lot.

Note by Aloysius Ng
5 years, 5 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


Sort by:

Top Newest

It can be done.

3n=20(n0)+21(n1)+22(n2)++2n1(nn1)+2n(nn){ 3 }^{ n }={ 2 }^{ 0 }\left( \begin{matrix} n \\ 0 \end{matrix} \right) +{ 2 }^{ 1 }\left( \begin{matrix} n \\ 1 \end{matrix} \right) +{ 2 }^{ 2 }\left( \begin{matrix} n \\ 2 \end{matrix} \right) +\cdots +{ 2 }^{ n-1 }\left( \begin{matrix} n \\ n-1 \end{matrix} \right) +2^{ n }\left( \begin{matrix} n \\ n \end{matrix} \right)

Joel Yip - 5 years, 4 months ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...