Series (Edexcel International A Level (IAL) Further Maths: Further Pure 1): Exam Questions

In this question use the standard results for summations.

Show that for all positive integers $n$

$\sum _{r=1}^n(12r^2+2r-3)=A{n}^3+B{n}^2$

where $A$and $B$ are integers to be determined.

Hence determine the value of $n$ for which

$\sum _{r=1}^{2n}{r}^3-\sum _{r=1}^n(12r^2+2r-3)=270$

Use the standard results for summations to show that, for all positive integers $n$,

$\sum _{r=1}^{n}r\left(2r^2-3r-1\right)=\frac{1}{2}n(n+1{)}^2(n-2)$

Hence show that, for all positive integers $n$,

$\sum _{r=n}^{2n}r\left(2r^2-3r-1\right)=\frac{1}{2}n(n-1)(an+b)(cn+d)$

where $a$, $b$, $c$ and $d$ are integers to be determined.

Use the standard results for $\sum _{r=1}^n{r}^2$ and $\sum _{r=1}^n{r}^3$ to show that, for all positive integers $n$

$\sum _{r=1}^n{r}^2(r+2)=\frac{1}{12}n(n+1)(a{n}^2+bn+c)$

where $a$, $b$ and $c$ are integers to be determined.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Use the standard results for $\sum _{r=1}^{n}r$ and $\sum _{r=1}^n{r}^2$ to show that for all positive integers $n$

$\sum _{r=1}^n(7r-5{)}^2=\frac{n}{6}(7n+1\left)\right(An+B)$

where $A$and $B$ are integers to be determined.

Use the standard results for $\sum _{r=1}^n{r}^2$ and $\sum _{r=1}^{n}r$ to show that for all positive integers $n$

$\sum _{r=0}^n(r+1)(r+2)=\frac{1}{3}(n+1)(n+2)(n+3)$

Hence determine the value of

$10\times 11+11\times 12+12\times 13+\dots +100\times 101$

Prove by induction that, for $n\in \mathrm{\mathbb{N}}$

$\sum _{r=1}^n{r}^3=\frac{1}{4}{n}^2(n+1{)}^2$

Using the standard summation formulae, show that

$\sum _{r=1}^{n}r(r+1)(r-1)=\frac{1}{4}n(n+A)(n+B)(n+C)$

where $A$, $B$ and $C$ are constants to be determined.

Determine the value of $n$ for which

$3\sum _{r=1}^{n}r(r+1)(r-1)=17\sum _{r=n}^{2n}{r}^2$

Use the standard results for $\sum _{r=1}^n{r}^3,\sum _{r=1}^n{r}^2\text{and}\sum _{r=1}^{n}r$ to show that for all positive integers $n$,

$\sum _{r=1}^{n}r(r-1)(r-3)=\frac{1}{12}n(n+1)(n-1)(3n-10)$

Hence show that

$\sum _{r=n+1}^{2n+1}r(r-1)(r-3)=\frac{1}{12}n(n+1)(a{n}^2+bn+c)$

where $a$, $b$ and $c$ are integers to be determined.

Prove by induction that for $n\in \mathrm{\mathbb{N}}$

$\sum _{r=1}^n{r}^2=\frac{n}{6}(n+1)(2n+1)$

Hence show that

$\sum _{r=1}^n(r^2+2)=\frac{n}{6}(a{n}^2+bn+c)$

where $a$, $b$ and $c$ are integers to be found.

Using your answers to part (b), find the value of

$\sum _{r=10}^{25}(r^2+2)$

Using the formulae for $\sum _{r=1}^{n}r\text{and}\sum _{r=1}^n{r}^2$, show that

$\sum _{r=1}^n(r+1)(r+5)=\frac{n}{6}(n+7)(2n+7)$

for all positive integers $n$.

Hence show that

$\sum _{r=n+1}^{2n}(r+1)(r+5)=\frac{7n}{6}(n+1)(an+b)$

where $a$ and $b$ are integers to be determined.

Use the standard results for $\sum _{r=1}^n{r}^2$ and $\sum _{r=1}^{n}r$ to show that

$\sum _{r=1}^n(2r-1{)}^2=\frac{1}{3}n(4n^2-1)$

for all positive integers $n$.

Hence find the exact value of the sum of the squares of the odd numbers between 200 and 500.