Please solve q 8?

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1 Answer
Feb 17, 2018

Answer is option #(4)#

Explanation:

General term, #T_n = 1/( (2n-1)^(1/2)+(2n+1)^(1/2))#

Rationalize the General term of the sequence :-

#rArr T_n = ( (2n-1)^(1/2)-(2n+1)^(1/2)) /[( (2n-1)^(1/2)+(2n+1)^(1/2)).( (2n-1)^(1/2)-(2n+1)^(1/2))]#

#rArr T_n = ((2n-1)^(1/2)-(2n+1)^(1/2))/(-2)#

#rArr T_n = ((2n+1)^(1/2)-(2n-1)^(1/2))/2#

Now summing all the terms ie.

#rArr sum T_n = sum_(n=0)^n((2n+1)^(1/2)-(2n-1)^(1/2))/2#

#rArr sum T_n = [(3-1)+(5-3)+.............+((2n+1)^(1/2)-(2n-1)^(1/2))]/2#

#rArr sum T_n = [(cancel3-1)+(cancel5-cancel3)+.............+((2n+1)^(1/2)-cancel(2n-1)^(1/2))]/2#...........................#($)#

#:. sum T_n = [(2n+1)^(1/2)-1]/2#....................option #(4)#

#NOTE# :- In equation #($)# we should note a Pattern of

cancellation of the terms has formed ; where the initial

terms are eliminated by the succeeding terms at some point ,

except the Least and the Biggest terms which are #1# and

#(2n+1)^(1/2)# respectively and the term #(2n-1)^(1/2)# also gets

cancelled.