Find the value of 80402 * S?

Question

S = 1/(1 + 1^2 + 1^4)
+ 2/(1 + 2^2 + 2^4)
+ 3/(1 + 3^2 + 3^4)
+ 4/(1 + 4^2 + 4^4)
+ ..... .....
+ 200/(1 + 200^2 + 200^4).


Find the value of 80402 * S.



This question came from a local mathematical olympiad competition I had just a few weeks ago. I have finished it already but I am unsure of the answers. Note that no calculators are allowed, and also the answer is in integer form 0 < N <= 99999.

Please show how you do it.

Mona C,

thanks for answering! But the answer is an integer. Meaning 80402 * S is an integer (not rounded off too, exact answer). I didn't manage to get any answer so I left it blank.

I wrote a C program:


#include
main() {
double ans=0, count=0;
for (count=1;count <= 200;count++) {
ans = ans + 80402*count/(1 + count*count+count*count*count*count);
}
printf("%.3f", ans);
}



Output was "40200.000"

Can anyone explain that?

Sorry the line of code that got truncated off:

ans = ans + 80402 * count / (1 + count * count + count * count * count * count );

Mona C,

if you had answered 40201 then you are 1 more than the correct answer, which is wrong! lol. But we can deduce S is approx 0.5, but how can we arrive to that, and why is S = 40200/80402?

Answer 1

Hi, this is a pretty cool question. It seems to me that the sum can be expressed by

S(n) = n(n+1) / (2n(n+1)+2)

For example,

S(3) = 1/(1 + 1^2 + 1^4) + 2/(1 + 2^2 + 2^4)
+ 3/(1 + 3^2 + 3^4) = 3(4) / (2(3)(4) + 2) = 6/13

Similarly,

S(200) = 200(201) / (2(200)(201)+2) = 40200 / 80402

Unfortunately, what I have found is based on guesswork and some intuition. I'll probably need some time to figure out a more formal way to prove this. Alternatively, one can show via induction, but I'm guessing there's a more elegant way to do this. Probably after Monday when I finish my exam.

Answer 2

I calculate 'S' with microsoft excel, and the result is 0.49.

So 80402 * 0.49 = 39396.98

How? Is this ur answer?

Or if I rounded up the result of 'S' to be 0.5,

80402 * 0.5 = 40201.......

Answer 3

very nice question. I will think about it later and see if i come up with anything.

Answer 4

http://www.mathfiles.com/

Answer 5

i suck at math cant help ya