Which relation is true for A ?

Question

If A = $1/(sqrt1 + sqrt3)$ + $1/(sqrt3 + sqrt5)$ + $1/(sqrt5 + sqrt7)$ + ...... upto 50 terms, then which of the following relation is true for A :

4 $<=$ A $<=$ 4.5

9 $<=$ A $<=$ 10

4.5 $<=$ A $<=$ 5

None of the above

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Answer 1 · 2018-07-20T11:27:36

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

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

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 $(3) 4.5 le A le 5$ .