General Sequences & Series (Edexcel A Level Maths: Pure): Exam Questions

A sequence of terms $a_1,a_2,a_3,...$ is defined by

$a_1=3$

$a_{n+1}=8-a_n$

(i) Show that this sequence is periodic.

(ii) State the order of this periodic sequence.

Find the value of

$\sum _{n=1}^{85}{a}_n$

A sequence $u_1,u_2,u_3,...$ is defined by

$u_{n+1}=\frac{1}{u_n},u_1=\frac{2}{3}$

Find the exact value of $\sum _{r=1}^{100}{u}_r$.

The sequence $u_1,u_2,u_3,...$ is defined by

$u_{n+1}=k-\frac{24}{u_n}{u}_1=2$

where $k$ is an integer.

Given that $u_1+2u_2+u_3=0$, show that

$3k^2-58k+240=0$

Find the value of $k$, giving a reason for your answer.

Find the value of $u_3$.

A sequence of numbers $a_1,a_2,a_3,...$ is defined by

$a_{n+1}=\frac{k\left(a_n+2\right)}{a_n}n\in \mathrm{\mathbb{N}}$

where $k$ is a constant.

Given that

• the sequence is a periodic sequence of order 3

• $a_1=2$

show that

$k^2+k-2=0$

For this sequence explain why $k\ne 1$.

Find the value of

$\sum _{r=1}^{80}{a}_r$

A sequence $u_1,u_2,u_3,...$ is defined by

$\begin{array}{rcl}{u}_1& =& 23\\ u_{n+1}& =& u_n-3\end{array}$

Find the value of

     $\sum _{n=1}^{10}{u}_n$

Find the value of

$\sum _{n=11}^{15}{u}_n$

A sequence $u_1,u_2,u_3,...$ is defined by

$\begin{array}{rcl}{u}_1& =& 54\\ u_{n+1}& =& \frac{u_n}{3}\end{array}$

Find the exact value of

      $\sum _{n=1}^9{u}_n$

Find the exact value of

$\sum _{n=10}^{\infty }{u}_n$

A sequence $u_1,u_2,u_3,...$ is defined by

$u_k=k^2$

 for all $k\in \mathrm{\mathbb{N}}$.

State whether this sequence is increasing, decreasing, or neither.

A formula is given by

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

Use the formula to find the value of

$\sum _{k=1}^{50}{u}_k$

where $u_k$ is the sequence in part (a).

Find the sum of the squares of all the integers between 51 and 100 inclusive,

${51}^2+{52}^2+{53}^2+\dots +{99}^2+{100}^2$

A sequence $u_1,u_2,u_3,...$ is defined by

$\begin{array}{rcl}{u}_1& =& 2\\ u_{n+1}& =& p{u}_n-2\end{array}$

where $p$ is a constant.

Find expressions for $u_2$ and  $u_3$ in terms of $p$.

The sequence is periodic with order 2.

Find the value of $p$.

Using the value of $p$ in part (b), find

$\sum _{n=1}^{1001}{u}_n$

A sequence $u_1,u_2,u_3,...$ is defined by $u_n=7+5n$ where $n\in \mathrm{\mathbb{N}}$.

Describe this sequence.

Given that $\sum _{n=1}^k{u}_n=1190$

(i) Show that $\left(5k+119\right)\left(k-20\right)=0$

(ii) Hence find the value of $k$.

A sequence $u_1,u_2,u_3,...$ is defined by $u_n=5\times 2^n$ where $n\in \mathrm{\mathbb{N}}$.

Describe this sequence.

Given that 

$\sum _{n=1}^{k}5\times 2^n=20470$

find the value of $k$.

Use the value of $k$ in part (b) to find

$\sum _{n=1}^{k+3}5\times 2^n$

A sequence $u_1,u_2,u_3,...$ is defined by

$\begin{array}{rcl}{u}_1& =& 23\\ u_{n+1}& =& u_n+7\end{array}$

Find the value of

$\sum _{n=15}^{25}{u}_n$

Find the value of

$\sum _{n=1}^{25}\left(u_n-3\right)$

A sequence $u_1,u_2,u_3...$ is defined by

$\begin{array}{rcl}{u}_1& =& 35\\ u_{n+1}& =& u_n+7\cos \left(\frac{n\pi }{2}\right)-5{\left(-1\right)}^n\end{array}$

(i) Show that $u_2=40$

(ii) Find the value of $u_3$ and the value of $u_4$

Given that the sequence is periodic with order 4

(i) write down the value of $u_5$

(ii) find the value of $\sum _{r=1}^{25}{u}_r$

Show that

$\sum _{n=2}^{\infty }{\left(\frac{3}{4}\right)}^n\cos \left(180n\right)°=\frac{9}{28}$

Find the value of

$\sum _{r=4}^{\infty }20\times {\left(\frac{1}{2}\right)}^r$

Given that $\sum _{r=1}^k\left(31-6r\right)=-943$

(i) Show that $(3k+41)(k-23)=0$

(ii) Hence, find the value of $k$.

Given that

$\sum _{r=1}^{k}7\times 3^r=620004$

Find the value of $k$.

Use the value of $k$ in part (a) to find

$\sum _{r=0}^{k+3}7\times 3^r$

A sequence $u_1,u_2,u_3,...$ is defined by

$\begin{array}{rcl}{u}_1& =& 686\\ u_{n+1}& =& \frac{2u_n}{7}\end{array}$

Find the exact value of

$\sum _{n=7}^{\infty }{u}_n$

Find the value of

$\sum _{n=1}^{\infty }{u}_{n+4}$

Given that 

$\sum _{r=1}^k\left(89-5r\right)=-35$

find the value of $k$.

Given that

$\sum _{r=1}^{k}3\times {\left(-2\right)}^r=-262146$

find the value of $k$.

Use the value of $k$ in part (a) to find

$\sum _{r=5}^{k+2}3\times {\left(-2\right)}^r$

A sequence $u_1,u_2,u_3,...$ is defined by

$\begin{array}{rcl}{u}_1& =& 3\\ u_{n+1}& =& \left(p-2\right)u_n-2\end{array}$

where $p$ is a constant.

Given that the sequence is periodic with order 2, find the value of $p$.

Use the value of $p$ in part (a) to find

$\sum _{n=50}^{900}{u}_n$

Show that

$\sum _{n=1}^{48}{\log }_5\left(\frac{n+2}{n+1}\right)=2$

Show that $\sum _{r=1}^{16}\left(3+5r+2^r\right)=131798$

Given that 

$\sum _{n=2}^{\infty }\sqrt{3}\times {\left(\sqrt{2x}\right)}^{n-1}=3\sqrt{3}$

find the value of $x$.

Given that

$\sum _{n=7}^{12}\left(a+\left(n-1\right)d\right)=-69$

  $\sum _{n=7}^{16}\left(a+\left(n-1\right)d\right)=-175$

  $\sum _{n=1}^6\left(a+\left(n-1\right)d\right)=-13d$

find the values of $a$ and $d$.

A sequence $u_1,u_2,u_3,...$ is defined by

$u_n=\sqrt{13}+(-2{)}^{n-1}$

where $n\in \mathrm{\mathbb{N}}$.

Find the exact value of

$\sum _{n=11}^{23}{u}_n$

A sequence $u_1,u_2,u_3,...$ is defined by

         $u_k=-2k\times \left(\cos \right(k\pi ){)}^{k+1}$

where $k\in \mathrm{\mathbb{N}}$.

Determine, with reason, whether the sequence is increasing, decreasing, or neither.

A different sequence $v_1,v_2,v_3,...$ is defined by 

$v_k=\sin \left(kq\pi \right)$

where $k\in \mathrm{\mathbb{N}}$ and $q$ is a real number constant.

Given that the sequence is not periodic, suggest a possible value for $q$, giving a reason for your answer.

A sequence is defined by the recurrence relation  

$u_1=a$

$u_2=b$

$u_{k+2}=\frac{u_{k+1}}{u_k}$

where $a$ and $b$ are real numbers.

Show that the sequence is periodic, and find its order.

Given that 

$\sum _{r=1}^{44}{u}_r=-50$

$\sum _{r=1}^{84}{u}_r=-92$

find the possible values of $a$ and $b$.

Show that, for all $n\in \mathrm{\mathbb{N}}$,

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