Challenge?

Question

f(m, n) is defined for all positive integers m and n and satisfies
i. f(m, m) = m
ii. f(m, n) = f(n, m)
iii. f(m, m+n) = f(m, n) * (1 + m / n)
Find f(14, 52).

Answer 1

Nagpapatawa ba kayo?

364

Answer 2

f(14,52) can be written as f(14, 14+38), so using rule 3, f(14, 52) = f(14, 38) * (1 + 14/38). Of course, now we have to find f(14, 38), so using the same rule, f(14, 38) = f(14, 14+24) = f(14, 24) * (1 + 14/24). Now we find f(14, 24) = f(14, 10) * (1+14/10). Note that we cannot blindly substitute 14 and -4 into this forumla to find the next number because the function is only defined for positive integers. However, we can use rule 2 to find that f(14, 10) = f(10, 14) = f(10, 4) * (1+10/4). Now f(10, 4) = f(4, 10) = f(4, 6) * (1 + 4/6). f(4, 6) = f(4, 2) * (1+4/2). f(4,2) = f(2, 4) = f(2, 2) * (1+ 2/2). Finally, by rule 1, f(2,2) = 2 ad we can end the recursion. Back-substituting the appropriate values:

f(14,52) = 2 * (1+2/2) * (1+4/2) * (1+4/6) * (1+10/4) * (1+14/10) * (1+14/24) * (1+14/38) = 2 * 2 * 3 * 5/3 * 7/2 * 12/5 * 19/12 * 26/19 = 2489760 / 6840 = 364

So f(14, 52)=364

Answer 3

Iteratively applying f(m, m+n) = f(m, n) * (1 + m / n)

Solve,

f(14,52)

= f(14,14+38)

= f(14, 38)*(1+14/38)

= f(14,14+24)*(1+14/38)

= f(14,24)*(1+14/24)(1+14/38)

= f(14,14+10)*(1+14/24)(1+14/38)

= f(14,10)*(1+14/10)(1+14/24)(1+14/38)

Note f(14,10)=f(10,14)

= f(10,10+4)*(1+14/10)(1+14/24)(1+14/38)

= f(10,4)*(1+10/4)(1+14/10)(1+14/24)(1+14/38)

= f(4,4+6)*(1+10/4)(1+14/10)(1+14/24)(1+14/38)

= f(4,6)*(1+4/6)(1+10/4)(1+14/10)(1+14/24)(1+14/38)

= f(4,4+2)*(1+4/6)(1+10/4)(1+14/10)(1+14/24)(1+14/38)

= f(4,4)*(1+4/2)(1+4/6)(1+10/4)(1+14/10)(1+14/24)(1+14/38)

= 4(1+4/2)(1+4/6)(1+10/4)(1+14/10)(1+14/24)(1+14/38)

= 4(3)(1+2/3)(1+5/2)(1+7/5)(1+7/12)(1+7/19)

= 364

Answer 4

364

Answer 5

Based on given information(and the definition of f), f(14,52) is undefined.

^_^