Number theory problem of log

Question

!!1/(log_2\4) + 1/(log_4\4) + 1/(log_8\4) +.....+ 1/(log_(2^n)\4)!!

!!a)" "(n(n+1))/2!!

!!b)" "2/(n(n+1))!!

!!c)" "(n(n+1))/4!!

!!d)" "!!None

Solution

Let us first try to simplify logarithmic term.

!!1/(log_2\4) + 1/(log_4\4) + 1/(log_8\4) +.....+ 1/(log_(2^n)\4)!!

We know that !!log_a\b = 1/log_b\a!!

!!=log_4\2 + log_4\4 + log_4\8 +...........+ log_4(2^n)!!

!!=log_4\2 + log_4(2^2) + log_4(2^3) +...........+ log_4(2^n)!!

!!=log_4\2 + 2log_4\2 + 3log_4\2 +...........+ nlog_4\2!!

!!=log_4\2 (1+2+3+...........+n)!!

!!= 1/(log_2\4)xx(n(n+1))/2!!

!!= 1/(2(log_2\2))xx(n(n+1))/2!!

!!= (1/2)xx(n(n+1))/2!!

!!= (n(n+1))/4!!

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