Concept of series

Question

!!1,4,10,22,46,..... n!! terms. !!10^(th)!! term of series is?

(a) !!T_(10) < 1000!!

(b) !!T_(10) > 1500!!

(c) !!1000< T_(10)< 1500!!

(d) None

Solution

Series,

!!1,4,10,22,46,....n!! terms

!!T_1 =1 and T_2 = 4!!. Difference between !!T_1!! and !!T_2!! is !!3= 3xx2^0!!. Similarly between !!T_2!! and !!T_3!! is !!6= 3xx2^1!!.

So, we can say that,
!!T_(n)-T_(n-1)= 3xx2^(n-2)!!

!!=> T_n = T_(1)+ 3 sum_(i=0)^(n-2) 2^i!!

Put !!n=10!!

!!T_(10) = 1+ 3 sum_(i=0)^(n-2)2^i!!

!!=> T_(10) = 1+ 3[2^0+2^1+.......+2^8]!!

!!=> T_(10) = 1+ 3[(2^9-1)/(2-1)]= 1+ 3xx511 !!

!!=> T_10 = 1534!!

Therefore !!T_10 > 1500!!. So, option (b) is correct .