Question

In the pattern ${\displaystyle \frac{1}{3}+\frac{1}{4}+\frac{1}{5}+}$ ..... which fraction makes the sum greater than 1 (first time)? Explain.

Answer 1

${\displaystyle \frac{1}{3}+\frac{1}{4}+\frac{1}{5}=\frac{20+15+12}{60}}$

= ${\displaystyle \frac{47}{60}<1}$  ......[∵ Numerator < Denominator]

According to the pattern, next number will be ${\displaystyle \frac{1}{6}}$

∴ ${\displaystyle \frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}=\frac{40+30+24+20}{120}}$

= ${\displaystyle \frac{114}{120}<1}$  .....[∵ Numerator < Denominator]

Now, according to the pattern, next number after ${\displaystyle \frac{1}{6}}$ is ${\displaystyle \frac{1}{7}}$

∴ ${\displaystyle \frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}=\frac{280+210+168+140+120}{840}}$

= ${\displaystyle \frac{918}{840}>1}$   .....[∵ Numerator > Denominator]

Hence, ${\displaystyle \frac{1}{7}}$ makes the sum greater than 1 (first time).