<prompt> HebuSsqrt1dfrac112dfrac122sqrt1dfrac122dfrac132 cdotssqrt1dfrac120072dfrac120082.Kokotoa floor S2floor.  <response>: Kwa nukuu ya muhtasari, S sum_i12007 sqrt1 tfrac1i2 tfrac1i12 . Kwa kutumia dhehebu la kawaida na kurahisisha, tunaanzaalign S sum_i12007 sqrt fraci2 i2 2i 1 i2 2i 1 i2 i2 i12 sum_i12007 sqrt fraci4 2i3 3i2 2i 1 i2 i12 sum_i12007 sqrt fraci2 i 12i 2 i12 sum_i12007 fraci2 i 1i2 i sum_ i12007 1 frac1ii1 sum_i12007 1 frac1i - frac1i1 endalignAngalia sehemu hiyo ya maneno darubini, ikifanya hesabu kuwa rahisi. Matokeo ya hesabu kwa S 2007 1 - tfrac12008. Kwa hivyo, S2 2008 - tfrac120082 4032064 - 2 tfrac120082. Kwa kuwa 0 tfrac120082 1, tunahitimisha kuwa floor S2floor 4032062. 