Question

If $P={\sum }_{r=3n}^{r=4n-1}\left[\frac{r^2+13n^2-7rn}{n^3}\right]$
and $Q={\sum }_{r=3n+1}^{r=4n}\left[\frac{r^2+13n^2-7rn}{n^3}\right]$ then

• A. $P>\frac{5}{6}$
• B. $Q<\frac{5}{6}$
• C. $Q>\frac{5}{6}$
• D. $P<\frac{5}{6}$

Answer 1

Answer: (A), (C)

The correct options are
A $P>\frac{5}{6}$
C $Q>\frac{5}{6}$

Using Reimann Sum we get the following function.
$f\left(x\right)=x^2-7x+13$

For P $\to$

symmetric about $x=\frac{7}{2}$
here gain(extra) Area is greater than less area

For Q $\to$

So $P,Q>\frac{5}{6}$ where $\frac{5}{6}$ is the value of integral of $f\left(x\right)=x^2-7x+13$ from 3 to 4.