There are 25 stones in a heap. The heap is divided into two parts, then one of the parts is divided in two again, et cetera, until we have 25 separate stones. After each division of one of the heaps into two smaller heaps we write the product of the numbers of stones in these two heaps on a blackboard. Prove that at the end the sum of all the numbers on the blackboard is 300.
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1Seems like a dupe. Note that for $n=25$, $n(n-1)/2=300$. – cosmo5 Feb 03 '21 at 18:39
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O my gawd , thanks a bunch @cosmo5!!!!! I really appreciate it !! – Sukhoi Feb 03 '21 at 18:50
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Hint Prove by strong induction that if you have $n$ stones, at the end you get $$\frac{n(n-1)}{2}$$
The inductive step is easy: if you have $n+1$ stones and at the first step you divide them into $k$ and $n+1-k$ then the first number you get is $$k(n+1-k)$$
Next, apply $P(k)$ and $P(n+1-k)$.
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