Magical sequence

Algebra Level 5

The sequence ( a n ) (a_n) satisfies a 0 = 0 a_0=0 and a n + 1 = 8 5 a n + 6 5 4 n − a n 2 a_{n + 1} = \dfrac85a_n + \dfrac65\sqrt {4^n - a_n^2} for n ≥ 0 n\geq 0 . Find ⌊ ( a 10 ) ⌋ \lfloor{(a_{10})}\rfloor


The answer is 983.

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Parth Lohomi by Parth Lohomi , 20, India

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2 solutions

Pranjal Jain
Feb 5, 2015 
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#include<iostream.h>
#include<conio.h>
#include<math.h>

int main()
{
clrscr();
double a=0;
int n,c;
for (n=0;n<11;n++)
{
cout<<a<<"\n";
a=1.6*a+1.2*sqrt(pow(4.0,n)-a*a);
}
getch();
return 0;
}

Sorry for using programming! 😜

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Programming in algebric problem,....?

Parth Lohomi - 6 years, 4 months ago

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Yeah! I was just craving to use C++ program! Here I got that chance!

Pranjal Jain - 6 years, 4 months ago

Can you check disputes on JEE Mechanics 1?

Pranjal Jain - 6 years, 4 months ago

Can you post the algebraic solution? Thanks!

Calvin Lin Staff - 6 years, 4 months ago

Brilliant seemed to have died out for four years now! Isn't there a proper mathematical solution for this question? Or is it allowed to use a computer at a mathematics Olympiad?

William Steve Roy - 2 years, 5 months ago
汶良 林
Apr 22, 2015

a0 = 0

a1 = 6/5

a2 = 96/25

a3 = 936/125

a4 = 9600/625

a5 = 93600/3125

a6 = 960000/15625

a7 = 9360000/78125

a8 = 96000000/390625

a9 = 936000000/1953125

a10 = 9600000000/9765625

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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