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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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The first step of the VA (Variable Approach) method is to modify the original condition and the question. If we determine the value of f(x), then we can get a solution.
Condition 1)
Since f(a)= \(\sqrt{a}+\sqrt{a+1}\) ,
we have
1/f(a)=1/ \(\sqrt{a}+\sqrt{a+1}\)
= \(\frac{\left(\sqrt{a}-\sqrt{a+1}\right)}{\left(\sqrt{a}+\sqrt{a+1}\right)\left(\sqrt{a}-\sqrt{a+1}\right)}\) (multiplying both the denominator and numerator by the conjugate)
=\(\frac{\left(\sqrt{a}-\sqrt{a+1}\right)}{a-\left(a+1\right)}\) (multiplying the denominator)
= \(\frac{\left(\sqrt{a}-\sqrt{a+1}\right)}{\left(a-a-1\right)}\) (multiplying -1 through the bracket)
= \(\frac{\left(\sqrt{a}-\sqrt{a+1}\right)}{-1}\) (adding like terms)
= \(-\sqrt{a}+\sqrt{a+1}\) (dividing by -1)
Then 1/f(1)+1/f(2)+⋯+1/f(50)= \(\left(-\sqrt{1}+\sqrt{2}\right)+\left(-\sqrt{2}+\sqrt{3}\right)\) +⋯+\(\left(-\sqrt{50}+\sqrt{51}\right)=-1+\sqrt{51}\)
Since 7 < \(\sqrt{51}\) < 7.5, we have 6 < \(\sqrt{51}\-1 < 6.5 and the integer closest to r is 6.
Since condition 1) yields a unique solution, it is sufficient.
Condition 2)
Since we don’t have any specific definition of f, condition 2) does not yield a unique solution, and it is not sufficient.
Therefore, A is the answer.
Answer: A
