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

The first $k$ terms of a series are given by $\sum _{r=1}^k\left(7+5r\right)$.

Show that this is an arithmetic series, and determine its first term and common difference.

Given that   $\sum _{r=1}^k\left(7+5r\right)=1190$ ,

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

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

The first $k$ terms of a series are given by $\sum _{r=1}^{k}5\times 2^r$ .

Show that this is a geometric series, and determine its first term and common ratio.

Given that  $\sum _{r=1}^{k}5\times 2^r$ = 20470,

Show that $k=\frac{\log 2048}{\log 2}$

For this value of $k$, calculate $\sum _{r=1}^{k+3}5\times 2^r$.

A geometric series is given by $1+2x+4x^2+\dots$

(i) Write down the common ratio, $r$, of the series.

(ii) Given that the series is convergent, and that $\sum _{n=1}^{\infty }2x^{n-1}=19$, calculate the value of $x$.

An arithmetic series is given by $a+(a+d)+(a+2d)+\dots$

Given that $\sum _{n=1}^7\left(a+\left(n-1\right)d\right)=91$ and $\sum _{n=1}^{10}\left(a+\left(n-1\right)d\right)=175$, find the values of $a$ and $d$.

A sequence is defined for $k\ge 1$ by the recurrence relation  $u_{k+1}=u_k-3,u_1=23.$

Calculate

     $\sum _{n=1}^{10}{u}_n$

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

A sequence is defined for $k\ge 1$ by the recurrence relation $u_{k+1}=\frac{u_k}{3},u_1=54.$

Calculate, giving your answers as exact values 

      $\sum _{n=1}^9{u}_n$

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

A sequence is defined for $k\ge 1$by the recurrence relation $u_{k+1}=p{u}_k-2,u_1=2,$

where $p$ is a constant.

Write down expressions for $u_2$ and  $u_3$ in terms of $p$.

Given that the sequence is periodic with order 2, and given as well that $u_1\ne u_2$,

Find the value of $p$.

For the value of $p$ found in part (b)

Calculate $\sum _{n=1}^{1001}{u}_n$

The terms of a sequence are defined by $u_k=k^2$ for all $k\ge 1$.

State, with a reason, whether this sequence is increasing, decreasing, or neither.

It can be shown that, for all $n\ge 1$,

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

Using that formula,

Calculate $\sum _{r=1}^{50}{u}_r$

Find the value of  ${51}^2+{52}^2+{53}^2+\dots +{99}^2+{100}^2$,i.e. the sum of the squares of all the integers between 51 and 100 inclusive.

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  $\sum _{n=1}^9\left(a+\left(n-1\right)d\right)=-279$ and $\sum _{n=1}^{13}\left(a+\left(n-1\right)d\right)=-585$, find the values of $a$ and $d.$

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

Show that $k=\frac{\log 59049}{\log 3}$

For this value of $k$, calculate $\sum _{r=0}^{k+3}7\times 3^r$.

A convergent geometric series is given by $1-4x+16x^2-64x^3+\dots$

Write down the range of possible values of $x$.

Given that  $\sum _{n=1}^{\infty }{\left(-4x\right)}^{n-1}$=24

Calculate the value of $x$.

A sequence is defined for $k\ge 1$ by the recurrence relation  $u_{k+1}=u_k+7,u_1=23.$

Calculate

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

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

A sequence is defined for $k\ge 1$by the recurrence relation  $u_{k+1}=\frac{2u_k}{7},u_1=686.$

Calculate, giving your answers as exact values

$\sum _{n=7}^{\infty }{u}_n$ 

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

A sequence is defined for $k\ge 1$ by the recurrence relation $u_{k+1}=(p-2)u_k-2,u_1=3$

where $p$ is a constant.

Given that the sequence is periodic with order 2, and given as well that $u_1\ne u_2$,

Find the value of  $p$.

For the value of $p$ found in part (a),

Calculate  $\sum _{n=50}^{900}{u}_n$

The terms of a sequence are defined, for all $k\ge 1,$  by  $u_k=(-1{)}^k\times k^2$.

State, with a reason, whether this sequence is increasing, decreasing, or neither.

It can be shown that, for all $n\ge 1$,

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

Using those formulas,

Show that $\sum _{r=1}^{100}{u}_r=\sum _{r=1}^{100}r$.

Given that  $\sum _{r=1}^k\left(89-5r\right)=-35$, find the value of $k$.

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

(i) show that $\frac{k-1}{2}=\frac{\log 65536}{\log 4}$    

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

For this value of $k$, calculate $\sum _{r=5}^{k+2}3\times {\left(-2\right)}^r$.

Given that $\sum _{n=7}^{12}\left(a+\left(n-1\right)d\right)=-69$,  $\sum _{n=7}^{16}\left(a+\left(n-1\right)d\right)=-175$ and  $\sum _{n=1}^6\left(a+\left(n-1\right)d\right)=-13d$, find the values of $a$ and $d$.

A convergent geometric series is given by $\sqrt{3}+\sqrt{6x}+2x\sqrt{3}+\dots ,$, where in all cases the square root symbol indicates the positive square root of the number in question.

Write down the range of possible values of $x$.

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

Calculate the value of  $x$.

A sequence is defined for $k\ge 1$ by  $u_k=\sqrt{13}+(-2{)}^{k-1}$.

Calculate $\sum _{r=11}^{23}{u}_r$, giving your answer as an exact value.

A sequence is defined for all $k\ge 1$ by

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

Determine, giving reasons for your answer, whether the sequence is increasing, decreasing, or neither.

A different sequence is defined for all $k\ge 1$by $v_k=sin\left(kq\pi \right)$

where $q$ is a real constant.

Given that the sequence is not periodic,

suggest a possible value for $q$, giving a reason for your answer.

A sequence is defined for $k\ge 1$ by the recurrence relation  $u_{k+2}=\frac{u_{k+1}}{u_k},u_1=a,u_2=b$

where $a$ and $b$ are real numbers.

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

Given that $\sum _{r=1}^{44}{u}_r=-50$ and   $\sum _{r=1}^{84}{u}_r=-92$

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

Prove that, for all $n\ge 1$,

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