Challenging +1.8 This is a Further Maths question requiring students to relate a Riemann sum (rectangles) to a definite integral, then integrate a non-standard function involving a quadratic under a square root. The integration requires substitution or recognition of the derivative of sinh^(-1) or ln form, followed by algebraic manipulation to reach the given bound. While conceptually clear for Further Maths students, the technical execution across multiple steps and the non-routine integral make this significantly above average difficulty.
8
\includegraphics[max width=\textwidth, alt={}, center]{5b43cb39-7560-4484-ba6f-17303e986f47-10_369_1531_260_306}
The diagram shows the curve \(\mathrm { y } = \frac { 1 } { \sqrt { \mathrm { x } ^ { 2 } + \mathrm { x } + 1 } }\) for \(x \geqslant 0\), together with a set of \(n\) rectangles of unit width. By considering the sum of the areas of these rectangles, show that
$$\sum _ { r = 1 } ^ { n } \frac { 1 } { \sqrt { r ^ { 2 } + r + 1 } } < \ln \left( \frac { 1 } { 3 } + \frac { 2 } { 3 } n + \frac { 2 } { 3 } \sqrt { n ^ { 2 } + n + 1 } \right)$$
## Question 8:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Sum} = \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{7}} + \ldots + \frac{1}{\sqrt{n^2+n+1}}$ | M1 A1 | Forms sum of areas of rectangles; need to see consideration of areas, not just the sum given in the question |
| $< \int_0^n \frac{1}{\sqrt{x^2+x+1}}\,dx$ | M1 | Compares with integral. Limits have to be correct |
| $x^2 + x + 1 = \left(x+\frac{1}{2}\right)^2 + \frac{3}{4}$ | B1 | Completes the square |
| $\int_0^n \frac{1}{\sqrt{\left(x+\frac{1}{2}\right)^2+\frac{3}{4}}}\,dx = \left[\sinh^{-1}\!\left(\frac{x+\frac{1}{2}}{\frac{1}{2}\sqrt{3}}\right)\right]_0^n = \left[\sinh^{-1}\!\left(\frac{2x+1}{\sqrt{3}}\right)\right]_0^n$ | M1 A1 | Uses formula |
| $= \left[\ln\!\left(\frac{2x+1}{\sqrt{3}}+\sqrt{\frac{(2x+1)^2+3}{3}}\right)\right]_0^n = \left[\ln\!\left(\frac{2x+1+2\sqrt{x^2+x+1}}{\sqrt{3}}\right)\right]_0^n$ | M1 A1 | Uses logarithmic form of $\sinh^{-1}$ |
| $= \ln\!\left(\frac{2n+1+2\sqrt{n^2+n+1}}{\sqrt{3}}\right) - \ln\!\left(\frac{3}{\sqrt{3}}\right)$ | M1 | Inserts limits |
| $= \ln\!\left(\frac{1}{3}+\frac{2}{3}n+\frac{2}{3}\sqrt{n^2+n+1}\right)$ | A1 | AG |
8\\
\includegraphics[max width=\textwidth, alt={}, center]{5b43cb39-7560-4484-ba6f-17303e986f47-10_369_1531_260_306}
The diagram shows the curve $\mathrm { y } = \frac { 1 } { \sqrt { \mathrm { x } ^ { 2 } + \mathrm { x } + 1 } }$ for $x \geqslant 0$, together with a set of $n$ rectangles of unit width. By considering the sum of the areas of these rectangles, show that
$$\sum _ { r = 1 } ^ { n } \frac { 1 } { \sqrt { r ^ { 2 } + r + 1 } } < \ln \left( \frac { 1 } { 3 } + \frac { 2 } { 3 } n + \frac { 2 } { 3 } \sqrt { n ^ { 2 } + n + 1 } \right)$$
\hfill \mbox{\textit{CAIE Further Paper 2 2020 Q8 [10]}}