bagdbmba wrote:

Qs. \(1+\frac{1}{2}+\frac{1}{3}+.....+\frac{1}{16}\) is

A) between 1 and 2

B) between 2 and 3

C) between 3 and 4

D) between 4 and 5

E) between 5 and 6

Here is my solution for this one.

\(\begin{split}

A&=1+(\frac{1}{2}+\frac{1}{3})+(\frac{1}{4}+\frac{1}{5})+...+(\frac{1}{14}+\frac{1}{15})+\frac{1}{16}\\

&<1+(\frac{1}{2}+\frac{1}{2})+(\frac{1}{4}+\frac{1}{4})+...+(\frac{1}{14}+\frac{1}{14})+\frac{1}{16}\\

&=1+\frac{2}{2}+\frac{2}{4} +\frac{2}{6} +...+ \frac{2}{14}+\frac{1}{16}\\

&=1+1+(\frac{1}{2}+\frac{1}{3})+(\frac{1}{4}+\frac{1}{5})+(\frac{1}{6}+\frac{1}{7})+\frac{1}{16}\\

&<1+1+(\frac{1}{2}+\frac{1}{2})+(\frac{1}{4}+\frac{1}{4})+(\frac{1}{6}+\frac{1}{6})+\frac{1}{16}\\

&=2+\frac{2}{2}+\frac{2}{4}+\frac{2}{6}+\frac{1}{16}\\

&=2+1+\frac{1}{2}+\frac{1}{3}+\frac{1}{16}\\

&=3 + \frac{43}{48}<4

\end{split}\)

\(\begin{split}

A&=1+\frac{1}{2}+(\frac{1}{3}+\frac{1}{4})+...+(\frac{1}{15}+\frac{1}{16})\\

&>1+\frac{1}{2}+(\frac{1}{4}+\frac{1}{4})+...+(\frac{1}{16}+\frac{1}{16})\\

&=1+\frac{1}{2}+\frac{2}{4}+...+\frac{2}{14}+\frac{2}{16}\\

&=(1+\frac{1}{2}+\frac{1}{2})+(\frac{1}{3}+\frac{1}{4})+...+(\frac{1}{7}+\frac{1}{8})\\

&>2+(\frac{1}{4}+\frac{1}{4})+...+(\frac{1}{8}+\frac{1}{8})\\

&=2+\frac{2}{4}+\frac{2}{6}+\frac{2}{8}\\

&=2+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\\

&=2+\frac{13}{12}>3

\end{split}\)

Hence \(3 < A < 4\), the answer is C

The solution for this type of question requires a lot of time, so it isn't the best way to solve this one under 2 minutes during take actual GMAT test. We could solve for the general problem like this one:

\(B=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\)

also, calculating approximate value of B is rather tough.

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