stuck on a system of recurrence sequence

by Nonecludiangeofan, Mar 20, 2025, 10:32 PM

Please guys help me solve this nasty problem that i've been stuck for the past month:
Let \( (a_n) \) and \( (b_n) \) be two sequences defined by:
\[
a_{n+1} = \frac{1 + a_n + a_n b_n}{b_n} \quad \text{and} \quad b_{n+1} = \frac{1 + b_n + a_n b_n}{a_n}
\]for all \( n \ge 0 \), with initial values \( a_0 = 1 \) and \( b_0 = 2 \).

Prove that:
\[
a_{2024} < 5.
\]
(btw am still not comfortable with system of recurrence sequences)
This post has been edited 1 time. Last edited by Nonecludiangeofan, 3 hours ago

Number Theory

by MuradSafarli, Mar 20, 2025, 7:55 PM

find all natural numbers \( (a, b) \) such that the following equation holds:

\[
7^a + 1 = 2b^2
\]

Abelkonkurransen 2025 2a

by Lil_flip38, Mar 20, 2025, 11:10 AM

A teacher asks each of eleven pupils to write a positive integer with at most four digits, each on a separate yellow sticky note. Show that if all the numbers are different, the teacher can always submit two or more of the eleven stickers so that the average of the numbers on the selected notes are not an integer.

a^12+3^b=1788^c

by falantrng, Jul 8, 2024, 1:59 PM

Find all the natural numbers $a, b, c$ satisfying the following equation:
$$a^{12} + 3^b = 1788^c$$.

A huge group of children compare their heights

by Tintarn, Apr 22, 2024, 7:11 PM

$1000$ children, no two of the same height, lined up. Let us call a pair of different children $(a,b)$ good if between them there is no child whose height is greater than the height of one of $a$ and $b$, but less than the height of the other. What is the greatest number of good pairs that could be formed? (Here, $(a,b)$ and $(b,a)$ are considered the same pair.)
Proposed by I. Bogdanov

Numbers from 1 to 15 with rare properties

by hectorleo123, Jul 10, 2023, 10:17 PM

Let $a, b, c$ and $d$ be elements of the set $\{ 1, 2, 3,\ldots , 2014, 2015 \}$ such that $a < b < c < d$, $a + b$ is a divisor of $c + d$, and $a + c$ is a divisor of $b + d$. Determine the largest value that $a$ can take.
This post has been edited 1 time. Last edited by hectorleo123, Jul 10, 2023, 10:20 PM
L

Problem 5

by Functional_equation, Jun 6, 2020, 9:19 AM

$a,b,c$ are non-negative integers.
Solve: $a!+5^b=7^c$

Proposed by Serbia
This post has been edited 3 times. Last edited by Functional_equation, Sep 30, 2020, 8:03 AM

Pieces // Class of 2022 // Final Update On This Blog [Part 1/3]

by shiningsunnyday, Mar 29, 2018, 2:09 PM

Rejected from Harvard, Princeton, Yale, MIT, UCLA, UCSD.
Your spots are the most coveted of any US college.
Yet you took a chance in me when nobody else did.
This means the world to me.
Destiny calls from the West.

Committed Stanford Class of 2022!
Pieces from memory

Harvard
New kid
Betrayal
Isolation
Garbage
Class clown
A bit too far
Awakening
Hustle begins
Lonelier the higher you get
Hello, anyone there?
Overnight sensation
Back when I was nothing.
Where were you guys again?
When I had nothing and stood bluffing.
Where were you back then?

Crazy they say my aspirations then,
Change they ask of me to fit in there,
Inspirational they say my aspirations are now,
Now people change to be more like me here.

Cause one's success comes at others' costs,
With scarcity there is competition, that's the natural law,
Which filters our behaviors through selfish thoughts,
Scarcity makes us products o' the system of us all.
Attachments:
This post has been edited 13 times. Last edited by shiningsunnyday, Apr 1, 2018, 6:13 AM

Iran Inequality

by mathmatecS, Jun 11, 2015, 9:05 AM

When $x(\ge1),$ $y(\ge1),$ $z(\ge1)$ satisfy $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2,$ prove in equality.
$$\sqrt{x+y+z}\ge\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}$$
This post has been edited 1 time. Last edited by mathmatecS, Jun 11, 2015, 9:31 AM

Inequality by Po-Ru Loh

by v_Enhance, Dec 29, 2012, 12:39 AM

Let $x,y,z \ge 1$ be real numbers such that \[ \frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1. \] Prove that \[ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1. \]

Inequality involving x, y and z

by cefer, Apr 28, 2012, 9:15 AM

The ones who are crazy enough to think they can change the world are the ones who do.

avatar

shiningsunnyday
Archives
+ January 2018
Shouts
Submit
  • The blog is locked right?

    by First, Apr 14, 2018, 6:00 PM

  • Great, amazing, inspiring blog. Good luck in life, and just know I aspire to succeed as you will in the future.

    by mgrimalo, Apr 7, 2018, 6:19 PM

  • Yesyesyes

    by shiningsunnyday, Mar 29, 2018, 5:30 PM

  • :O a new background picture

    by MathAwesome123, Mar 29, 2018, 3:39 PM

  • did you get into MIT?

    by 15Pandabears, Mar 15, 2018, 10:42 PM

  • wait what new site?

    by yegkatie, Feb 11, 2018, 1:49 AM

  • Yea, doing a bit of cleaning before migrating to new site

    by shiningsunnyday, Jan 21, 2018, 2:43 PM

  • Were there posts made in December 2017 for this blog and then deleted?

    I ask because I was purging my thunderbird inbox and I found emails indicating new blog posts of yours.

    email do not lie

    by jonlin1000, Jan 21, 2018, 12:12 AM

  • @below sorry not accepting contribs

    by shiningsunnyday, Dec 11, 2017, 11:15 AM

  • contrib plez?
    also wow this blog is very popular

    by DavidUsa, Dec 10, 2017, 7:53 PM

  • @First: lol same

    first shout of december

    by coolmath34, Dec 6, 2017, 2:32 PM

  • XD this blog is hilarious

    by Mitsuku, Nov 21, 2017, 7:40 PM

  • @wu2481632: stop encouraging SSD to procrastinate(blog entries are fun but procrastination isn't).

    by First, Aug 7, 2017, 5:02 PM

  • 3.5 weeks without a post :o

    by Flash12, Aug 4, 2017, 8:10 AM

  • First august shout!!

    by adik7, Aug 1, 2017, 6:52 AM

416 shouts
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