Two naughty boys...

Level pending

Brenda has an infinite number of coins. Unfortunately, her 2 mischievous younger brothers, Brian and Billy, stole an increasing number of coins from her. The first day, Brian stole 1 coin from her, the second day, 2 , the third day, 3 and so on. Billy stole 1 coin on the second day, 2 coins on the fourth day, 3 coins on the sixth day and so on. Up to the nth day, Brenda lost a n 2 + b n c \frac{an^2 + bn}{c} coins (this expression is in its simplest form). a+b+c= ?

Assumptions: n is an even number.


The answer is 19.

Noel Lo by Noel Lo , 24, Singapore

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1 solution

If we assume that n n is even, then after n n days Brian will have stolen n ( n + 1 ) 2 \dfrac{n(n + 1)}{2} coins and Billy will have stolen n 2 ( n 2 + 1 ) 2 = n ( n + 2 ) 8 \dfrac{\frac{n}{2}(\frac{n}{2} + 1)}{2} = \dfrac{n(n + 2)}{8} coins. The total number of coins stolen from Brenda up to and including the n n th day will then be

4 n ( n + 1 ) 8 + n ( n + 2 ) 8 = 4 n 2 + 4 n + n 2 + 2 n 8 = 5 n 2 + 6 n 8 . \dfrac{4n(n + 1)}{8} + \dfrac{n(n + 2)}{8} = \dfrac{4n^{2} + 4n + n^{2} + 2n}{8} = \dfrac{5n^{2} + 6n}{8}.

Thus a + b + c = 5 + 6 + 8 = 19 . a + b + c = 5 + 6 + 8 = \boxed{19}.

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