If x denotes the greatest integer ≤x, then value of $$S=$$[$$1$$]$$+$$[$$2$$]$$+$$[$$3$$]$$+$$…$$+$$[$$2024$$] is

Question

If x denotes the greatest integer ≤x, then value of $$S=$$[$$1$$]$$+$$[$$2$$]$$+$$[$$3$$]$$+$$…$$+$$[$$2024$$] is

Infinity Learn · Accepted Answer

For any positive integer k and x, we have[x]$$=$$ k ⇔ $$k2$$≤x<(k+1)2$$⇔x∈$$k2$$,$$k2+2k$$∴        [x]$$=1$$ for $$x=1$$,$$2$$,$$3$$.           [x]$$=2$$ for $$x=4$$,$$5$$,$$6$$,$$7$$,$$8$$,           [x]$$=3$$  for  $$x=9$$, $$10$$, $$11$$, $$12$$, $$13$$, $$14$$, $$15$$,           . . . . . .         [x]$$=44$$  for  $$x=1936$$, ......., $$2024Thus$$,  $$S=(1)(3)+(2)(5)+3(7)+$$…$$+44(89)              $$=$$∑$$k=144$$ $$(k)(2k+1)=2$$∑$$k=144$$ $$(k)2+$$∑$$k=144$$ $$(k)=26(44)(45)(89)+12(44)(45)=59730$$

If $[x]$ denotes the greatest integer $\leq x,$ then value of $S = [\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + \dots + [\sqrt{2024}]$ is

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If x denotes the greatest integer ≤x, then value of S=[1]+[2]+[3]+…+[2024] is

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If $[x]$ denotes the greatest integer $\leq x,$ then value of $S = [\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + \dots + [\sqrt{2024}]$ is