Under construction Stations under construction may not yet contain a good range of resources covering all the key questions and different types of problem.

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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Resource type Title
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 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
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 is the sum of the multiples of 5 or 8 but not both? R8249
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