## What are some interesting sequences of numbers?

### Key questions

1. 1

What is the difference between a sequence and a series?

2. 2

What properties does an arithmetic progression have?

3. 3

What properties does a geometric progression have?

4. 4

How can we evaluate sums such as $\sum_{k=1}^n k$ and $\sum_{k=1}^n k^2$?

5. 5

When and how can we sum an infinite series?

6. 6

What does it mean for a sequence or series to converge or diverge?

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#### Introducing...

Resource type Title
Rich example Change one thing
Building blocks Sort it out
Food for thought Square spirals

#### Developing...

Resource type Title
Rich example Bouncing to nothing
Package of problems Common terms
Many ways problem Can you find... series edition
Food for thought Connect three?
Investigation Same or different?
Go and think about it... A puzzling pentagon
Bigger picture Achilles and the tortoise

#### Review questions

Title Ref
Can we find the sum of the integers from $2k$ to $4k$ inclusive? R5849
Can we show $L < \int_1^r \frac{1}{x} \:dx < R$? R5602
Can we sum $1^2-2^2+3^2-4^2+\dotsb+ (2n-1)^2-(2n)^2$? R9819
Can we sum from $1000$ to $2000$ excluding multiples of 5? R7424
Can we sum the first $2n$ terms of $1,1,2,\frac{1}{2},4,\frac{1}{4},8,\frac{1}{8},..$? R6257
Can we sum the first $2n$ terms of $1^2-3^2+5^2-7^2+\cdots$? R5416
Find an expression for the sum of $r^2$ R6143
Given $S_n = S_{3n}$, can we express $a$ in terms of $n$? R9613
Given the sequence $(x_n)$, what is $\sum_{k=0}^\infty 1/x_k$? R8248
How are the $p$th, $q$th and $r$th terms of an AP related? R6137
How are the sums of $n$, $2n$ and $3n$ terms connected here? R8493
How are these recursively defined sequences related? R6555
How do we add the odd-numbered terms of a geometric series? R8617
If $27$, $x$, $y$ are in GP, with sum $21$, what are $x$ and $y$? R7258
If $S_n = 6 - 2^{n+1}/3^{n-1}$, can we show that we have a GP? R8163
If $u_n = a + bn + c2^n$, what's the sum of the first $n$ terms? R6153
If $x_1 = (x_0 + 2)/(x_0 + 1)$, can we show $\sqrt{2}$ lies between $x_0$ and $x_1$? R8404
If $x_2 = 1/(1 - x_1), x_3 = 1/(1 - x_2)$, can we show $x_1x_2x_3+1=0$? R5995
If AP terms $a_4,a_8, a_{16}$ are in GP, are $a_3, a_6, a_{12}$ in GP too? R5345
If a GP has $S_n=(3^n - 2^n)/2^{n-5}$, what's its common ratio? R6608
If an AP has $S_{10}=3S_5$, what's the ratio of $u_{10}$ to $u_5$? R8907
Should I hire a TV or borrow to buy it? R9375
What can we say if $a, x_1, x_2, x_3, x_4, x_5, b$ are in arithmetic progression? R6405
What's the sum of the $r^{th}$ bracket in $(1), (2,3), (4,5,6),\cdots$? R7240
When does $1 + 2/3 + (2/3)^2 + \dotsb$ first exceed $0.9999S_\infty$? R9185
When does $1-2+3-4+5-\cdots +(-1)^{n+1}n$ reach $100$? R8998
When does the sum of this series equal $60$? R8121
When does the sum of this series first exceed $2999/4000$? R7487