Pages and sums of pages

Number Theory Level 3

Given a book whose first page is numbered "1", let the number of any one page be P n P_n and the sum of all the pages up to and including any one page be P s P_s . Then, given P n P_n , it follows that P s = P n 2 + P n 2 . P_s=\frac{P_n^{2}+P_n}{2}. Now, given P s , P_s, what equation describes P n ? P_n?

Bonus: Why is this?

P n = P s 2 + P s 2 P_n=\frac{P_s^{2} + P_s}{2} P n = 2 P s P_n=2\sqrt{P_s} P n = ⌊ 2 P s ⌋ P_n=\Big\lfloor\sqrt{2P_s}\Big\rfloor P n = ⌊ ( P s 2 + P s ) ⌋ P_n=\Big\lfloor\big(P_s^{2} + P_s\big)\Big\rfloor

David Stiff by David Stiff , 20, Canada

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1 solution

David Vreken
Dec 30, 2017

Observe that P n 2 < P n 2 + P n < P n 2 + 2 P n + 1 P_n^2 < P_n^2 + P_n < P_n^2 + 2P_n + 1 .
So P n 2 < 2 P s < ( P n + 1 ) 2 P_n^2 < 2P_s < (P_n + 1)^2 , or P n < 2 P S < P n + 1 P_n < \sqrt{2P_S} < P_n + 1 .
Therefore, P n = ⌊ 2 P S ⌋ P_n = \lfloor \sqrt{2P_S} \rfloor .

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