It's Not Magic

\[\dfrac{19}{81}\approx 0.\red{2}\blue{3}\red{4}\blue{5}\red{6}\blue{7}\red{9}\red{0}\blue{1}...\] \[\dfrac{199}{9801} \approx 0.\red{02}\blue{03}\red{04}\blue{05}\red{06}\blue{07}\red{08}\blue{09}\red{10}\blue{11}...\] \[How\space to\space generate\space such\space rationals?\] \[let \space there \space be \space a \space sequence \space a_n\] \[a_1=\alpha\beta\] \[\forall n\in\mathbb{N},a_{n+1}=(\alpha+nd)\beta r^n\] \[let \space S_n = \sum_{k=1}^{n}a_k\] \[=\sum_{k=1}^{n} (\alpha+(k-1)d)\beta r^{k-1}\] \[=\sum_{k=1}^{n} (\alpha\beta r^{k-1}+(k-1)d\beta r^{k-1})\] \[=\alpha\beta\sum_{k=1}^{n}r^{k-1}+\beta d\sum_{k=1}^{n}(k-1)r^{k-1}\] \[=\alpha\beta\dfrac{r^{n}-1}{r-1}-\beta d\sum_{k=1}^{n}r^{k-1}+\beta d\blue{\sum_{k=1}^{n}kr^{k-1}}\] \[=\beta\dfrac{r^{n}-1}{r-1}(\alpha-d)+\beta d\blue{S}\] \[\blue{S} =\sum_{k=1}^{n}kr^{k-1}\] \[S\times r=\sum_{k=1}^{n}kr^{k}=nr^n+\sum_{k=2}^{n}(k-1)r^{k-1}\] \[\Rightarrow S(1-r)=-nr^n+\sum_{k=1}^{n-1}r^k\] \[=\dfrac{r^n-1}{r-1}-nr^n\] \[\Rightarrow S=\dfrac{r^n-1}{(r-1)^2}-\dfrac{nr^n}{r-1}\] \[Putting \space this\space we\space get\] \[S_n=\dfrac{\beta(1-r^n)(\alpha(1-r)+dr)}{(1-r)^2}-\dfrac{\beta dnr^n}{1-r}\] \[\Rightarrow \sum_{k=1}^{\infty}a_k=\dfrac{\beta(\alpha(1-r)+dr)}{(1-r)^2}\] \[Put\space \alpha=1=\beta=d, r=10^{-n},n\in\mathbb{N}\] \[\Rightarrow \sum_{k=1}^{\infty}a_k=\dfrac{100^n}{(10^n-1)^2}\] \[Now\space consider B_n=\dfrac{100^n}{(10^n-1)^2}-1\] \[\Rightarrow \boxed{B_n=\dfrac{2\times 10^n - 1}{(10^n - 1)^2}}\] \[Which\space is \space the \space Formula \space we \space wanted!\]

Note by Zakir Husain
3 months, 3 weeks 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}

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...