Sequences & Series (Edexcel IAL Maths): Exam Questions

Calculate

$\sum_{r = 1}^{5} 2 r + 1$

The sum given in part [a] is an arithmetic series. Write down the first term and the common difference.

Calculate

$\sum_{r = 1}^{3} 2 ( 3 )^{r}$

The sum given in part [a] is a geometric series. Write down the first term and the common ratio.

It is given that

$\sum_{r = 1}^{4} a ( r + 2 ) = 72$

where is a positive integer.

(i) Show that $18a=72$.

(ii) Find the value of $a$.

Determine if the series is arithmetic or geometric, justifying your answer.

The $n^{th}$ term of an arithmetic series is given by $u_{n}=3n+5$. Write the sum of the series, up to the $n^{th}$ term, in sigma notation.

The $n^{th}$ term of a geometric series is given by $u_{n}=5\times2^{n-1}$. Write the sum of the series, up to the $n^{th}$ term, in sigma notation.

Given that

$\sum_{r = 1}^{k} r^{2} = 55$

determine the value of $k$.

The first $k$ terms of a series are given by $\sum_{r = 1}^{k} ( 7 + 5 r )$

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

Given that $\sum_{r = 1}^{k} ( 7 + 5 r )$ $=1190$,

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

(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 $k=\frac{\mathrm{log}2048}{\mathrm{log}2}$

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

A geometric series is given by $1+2x+4x^{2}+...$

Write down the common ratio, $r$ , of the series.

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

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

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

The terms of a sequence are defined by $u_{k}=k^{2}$ for all $k\geq1$.

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

It can be shown that, for all $n\geq1$,

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

Using that formula,

Calculate $\sum_{r = 1}^{50} u_{r}$

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

Given that $\sum_{r = 1}^{k} ( 31 - 6 r ) = - 943$,

(i)

Show that $(3k+41)(k-23)=0$

(ii)

Hence, find the value of $k$.

Given that $\sum_{n = 1}^{9} ( a + ( n - 1 ) d ) = - 279$ and $\sum_{n = 1}^{13} ( a + ( n - 1 ) d ) = - 585$, find the values of $a$ and $d$ .

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

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

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

A convergent geometric series is given by $1-4x+16x^{2}-64x^{3}+...$

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

Given that $\sum_{n = 1}^{ \infty } ( - 4 x )^{n - 1} = 24$

Calculate the value of $x$.

The terms of a sequence are defined, for all $k\geq1,$ 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 $n\geq1$,

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

Using those formulas,

Show that $\sum_{r = 1}^{100} u_{r} = \sum_{r = 1}^{100} r$.

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

Given that $\sum_{r = 1}^{k} 3 \times ( - 2 )^{r} = - 262146$,

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

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

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

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

A convergent geometric series is given by $\sqrt{3}+\sqrt{6x}+2x\sqrt{3}+...$, 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 $\sum_{n = 2}^{ \infty } \sqrt{3 \times ( \sqrt{2} x} )^{n - 1} = 3 \sqrt{3}$

Calculate the value of $x$.

A sequence is defined for $k\geq1$ by $u_{k} = \sqrt{13} + \left(- 2\right)^{k - 1}$ .

Calculate $\sum_{r = 11}^{23} u_{r}$ , giving your answer as an exact value.

A sequence is defined for all $k\geq1$ by

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

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

A different sequence is defined for all $k\geq1$ by

$v_{k}=\mathrm{sin}(kqπ)$

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.

Prove that, for all $n\geq1$,

$\sum_{r = 1}^{n} ( 2 r )^{2} - \sum_{r = 1}^{n} ( 2 r - 1 )^{2} = \sum_{r = 1}^{2 n} r$