Which relation is true for A ?

If A = 1/(sqrt1 + sqrt3)11+3 + 1/(sqrt3 + sqrt5)13+5 + 1/(sqrt5 + sqrt7)15+7 + ...... upto 50 terms, then which of the following relation is true for A :

  1. 4 <= A <= 4.5
  2. 9 <= A <= 10
  3. 4.5 <= A<= 5
  4. None of the above

1 Answer
Jul 20, 2018

"The right option is "(3) 4.5 le A le 5The right option is (3)4.5A5.

Explanation:

A=1/(sqrt3+sqrt1)+1/(sqrt5+sqrt3)+1/(sqrt7+sqrt5)+...50"terms".

If A_n is the n^(th) term, then, A_n=1/(sqrt(2n+1)+sqrt(2n-1)),

={sqrt(2n+1)-sqrt(2n-1)}/{(2n+1)-(2n-1)}.

rArr A_n=1/2{sqrt(2n+1)-sqrt(2n-1)}.

:. A=A_1+A_2+A_3+...+A_49+A_50,

=1/2{(cancelsqrt3-sqrt1)+(cancelsqrt5-cancelsqrt3)+(cancelsqrt7-cancelsqrt5)+...

+(cancelsqrt99-cancelsqrt97)+(sqrt101-cancelsqrt99)}.

:. 2A=sqrt101-1............(ast).

Now, 100 lt 101 lt 121.

:. sqrt100 lt sqrt101 lt sqrt121,.

or, 10 lt sqrt101 lt 11.

"Adding" -1," we get, "9 lt sqrt101-1 lt 10.

:. (ast ) rArr 9 lt 2A lt 10.

"Dividing by "2 gt 0," we get, "4.5 lt A lt 5.

Hence, the right option is (3) 4.5 le A le 5.