Let us define [x] as the greatest integer which is less than or equal to x and logp(q) is logarithm of q to the base p. Find a natural n such that
[log2(1)]+[log2(2)]+[log2(3)]+....+[log2(n)]=2008
a) 312 b) 313 c) 314 d) 315 e) none of these
1,2,4,8,16,32,64,128,256 are powers of 2 that we will encounter here...
ReplyDeleteno. of terms that give values that are integral, is nothing but the no. of terms between two consecutive powers of 2,meaning:
no.s from 1-2: give log2() as 0(not including last no.)
from 2-4: 1, 2 no.s-2,3
from 4-8: 2, 4 no.s--4,5,6,7
from 8-16: 3, 8 no.s:8,9,10,11,12,13,14,15
16-32: 4, 16 no.s
32-64: 5, 32 no.s, and from 64-128: 6, 64 no.s, and 128-256: 7, 128 no.s...
total is: 2+8+24+64+160+384+896=1538..so no. left is=2008-1538=470, which has 58 8's and a 6...this means 58 no.s after 256,i.e 313...any other no. will add 8, and sum exceeds 2008
so none of these should be the answer
very nicely done, thats correct !
ReplyDelete