Sequences: All Exercises
Arithmetic Sequences
Find the general term $u_n$ for an arithmetic sequence with $u_3=8$ and $u_8=-17$.
$u_n=$
Find $k$ given that $3k+1, k$, and $-3$ are consecutive terms of an arithmetic sequence.
$k=$
Insert three numbers between $4.8$ and $12$ to make an arithmetic sequence.
Separate the numbers with commas.
Separate the numbers with commas.
Find the two values of $k$ that make $5, k, k^2-8$ consecutive terms of an arithmetic sequence.
Separate your answers with a comma.
Separate your answers with a comma.
$k=$
How many terms are in the sequence $14, 8, 2, -4, ..., -244$
$\frac{1}{k}, k, k^2+1$ are the 3rd, 4th and 6th terms of an arithmetic sequence. Find $k$ where $k \in \mathbb Z$
$k=$
How many positive multiples of 3 are there in the sequence $500, 492, 484, 476, ...$?
Geometric Sequences
A geometric sequence has $u_1=32$ and $u_6=-1$. Find the 10th term.
$u_{10}=$
A geometric sequence has $u_2=-6$ and $u_5=162$. Find the general term.
Give your answer in the form $a\left(b\right)^{n-1}$ where $a, b$ are integers.
Give your answer in the form $a\left(b\right)^{n-1}$ where $a, b$ are integers.
$u_n=$
A geometric sequence has $u_1=3, r=\sqrt{2}$ and $u_k=48$. Find the value of $k$.
$k=$
How many terms are in the sequence $3, 2, \frac{4}{3}, ..., \frac{128}{729}$?
$k-1, 2k$, and $21-k$ are consecutive terms of a geometric sequence. Find the two values of $k$.
Separate your answers with a comma.
Separate your answers with a comma.
$k=$
How many terms in the sequence $4, 4\sqrt{3}, 12, 12\sqrt{3}, ...$ are less than $5000$?
The difference between the third and the second term of a geometric sequence is $6$ times the first term. Find all possible values of the common ratio.
Separate your answers with a comma.
Separate your answers with a comma.
$r=$
Three positive integers form a geometric sequence. The sum of the three terms is $21$ and the product of the three terms is $216$. Find the three terms.
Separate the terms with commas.
Separate the terms with commas.
Sigma Notation
Use your calculator to calculate the following sums.
On the TI-Nspire CX, press 
then 
$\displaystyle \sum_{n=1}^{50} 2n+1=$
$\displaystyle \sum_{n=1}^{100} n=$
$\displaystyle \sum_{n=1}^{24} 1-\frac{2}{3}n=$
$\displaystyle \sum_{n=1}^{15} (-3)^{n-1}=$
Express the following series using sigma notation.
$6+9+12+15+18+21+24$
$\displaystyle \sum_{n=1}^{7}$
$6-12+24-48+96$
Give your answer in the form $a\left(b\right)^{n-1}$
Give your answer in the form $a\left(b\right)^{n-1}$
$\displaystyle \sum_{n=1}^{5}$
$1+4+9+16+25+36+49+64+81+100$
$\displaystyle \sum_{n=1}^{10}$
$\displaystyle 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\cdots $
Give your answer in the form $a\left(b\right)^{n-1}$
Give your answer in the form $a\left(b\right)^{n-1}$
$\displaystyle \sum_{n=1}^{\infty}$
Arithmetic Series
Find the sum of $4+7+10+13+\cdots\;$ to $50$ terms.
$S_{50}=$
Find the sum of all multiples of $7$ between $100$ and $1000$.
An arithmetic sequence has first term $4$ and common difference $5$.
If the sum of the first $n$ terms is $2772$. Find $n$.
If the sum of the first $n$ terms is $2772$. Find $n$.
$n=$
Find the sum of $-6+1+8+15+\;\cdots\;+141$.
Find the sum of $157+148+139\;\cdots\;-329$.
Find the sum of the positive terms of $285, 274, 263,\;\cdots$
The sixth term of an arithmetic sequence is $17$, and the sum of the first $13$ terms is $0$.
Find the first term of the sequence.
Find the first term of the sequence.
$u_1=$
An arithmetic series has $S_4=17$ and $S_{12}=123$. Find $S_{17}$.
$S_{17}=$
The fourth term of an arithmetic series is $14$. The sum of the first six terms is $72$.
Find the sixth term.
Find the sixth term.
$u_6=$
What is the sum of all the 3 digit numbers that start or end with $5$.
The sum of an arithmetic series is given by $S_n=n^2+2n$. Find the general term.
$u_n=$
Geometric Series
Find the sum of $2+6+18+54+162+486+1458$.
Find the sum of $3+6+12+24+48+\cdots$ to $13$ terms.
Find the sum of $4+12+36+108+\;\cdots\;+8748$.
How many terms are needed for the sum of the geometric series $4 + 12 + 36 + 108 + \cdots$ to exceed $100 000$?
Find the sum of the geometric series that has sixth term $6250$, ninth term $50$, last term $2$.
Consider a geometric series where the sum of the first two terms is $12$ and the sum of the first four terms is $60$. One sequence is $4, 8, 16, 32,\;\ldots$
Write down the first four terms of another sequence.
Write down the first four terms of another sequence.
The sum of the first $n$ terms of a geometric series is given by $S_n = 5^n – 1$.
Find the sixth term of this series.
Find the sixth term of this series.
$u_{6}=$
The sum of the first two terms of a geometric series is $4$ and the sum of the first three terms is $-14$. Find the two possible values for the first term.
Separate your answers with a comma
Separate your answers with a comma
$u_1=$
Find $n$ given $\displaystyle \sum_{k=1}^n2\times5^{k-1}=39062$
$n=$
A rope of length $6$m is cut into three pieces whose lengths form a geometric sequence.
The longest piece is four times the shortest piece. Find the length of the shortest piece.
Give your answer as a fraction
The longest piece is four times the shortest piece. Find the length of the shortest piece.
Give your answer as a fraction
Josh puts £1000 into an account which earns 10% p.a. interest paid annually.
At the end of each year, Josh puts in another £1000 in the account.
Find the value of the account after 6 years. Round your answer to the nearest £
At the end of each year, Josh puts in another £1000 in the account.
Find the value of the account after 6 years. Round your answer to the nearest £
£
Infinite Geometric Series
Which can be simplified to $\displaystyle S_\infty=$
$\displaystyle \frac{u_1}{1-r}$ where $\lvert r\rvert \lt 1$
Find the value of $\displaystyle \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\cdots$
Find the value of $\displaystyle \frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}-\frac{1}{32}+\cdots$
*Give your answer as a fraction
*Give your answer as a fraction
Find the value of $\displaystyle 2+\frac{4}{3}+\frac{8}{9}+\frac{16}{27}+\frac{32}{81}+\cdots$
Evaluate $\displaystyle{\sum_{n=1}^\infty \frac{3}{4^n}}$
Write the value of $\displaystyle 0.12+0.0012+0.000012+0.00000012+\cdots$ as a fraction.
The sum of the first three terms of a convergent infinite geometric series is $37$. The sum to infinity of the series is $64$. Find the general term in the form $u_n=\displaystyle u_1 \left(r\right)^{n-1}$.
$u_n=$
The sum of an infinite geometric series is $S_\infty =32$ and $S_3=28$.
Find the general term in the form $u_n=\displaystyle u_1 \left(r\right)^{n-1}$.
Find the general term in the form $u_n=\displaystyle u_1 \left(r\right)^{n-1}$.
$u_n=$
For what values of $x$ does the geometric series $3-24x^3+192x^6-1536x^9+\cdots$ converge.
$\lvert x \rvert<$
A geometric series converges to $8$ and $u_2=-\frac{5}{2}$. Find the common ratio.
$r=$
The geometric series $a+ar+ar^2+\cdots$ has a sum of $7$, and the terms involving odd powers of $r$ have a sum of $3$. What is $a+r$?
$a+r=$