Infinite Sequences

Key Questions

• How do find the `n`th term in a sequence?

  It depends on the type of sequence.

  If the sequence is an arithmetic progression with first term `a_1`, then the terms will be of the form:

  `a_n = a_1 + (n-1)b`
  for some constant b.

  If the sequence is a geometric progression with first term `a_1`, then the terms will be of the form:

  `a_n = a_1 * r^(n-1)`
  for some constant `r`.

  There are also sequences where the next number is defined iteratively in terms of the previous 2 or more terms. An example of this would be the Fibonacci sequence:

  `0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,...`

  Each term is the sum of the two previous terms.

  The ratio of successive pairs of terms tends towards the golden ratio `phi = 1/2 + sqrt(5)/2 ~= 1.618034`

  The terms of the Fibonacci sequence are expressible by the formula:

  `F_n = (phi^n-(-phi)^-n)/sqrt(5)` (starting with `F_0 = 0`, `F_1 = 1`)

  In general an infinite sequence is any mapping from `NN -> S` for any set `S`. It can be defined in any way you like.

  Finite sequences are the same, except that they are mappings from a finite subset of `NN` consisting of those numbers less than some fixed limit, e.g. `{n in NN: n <= 10}`

  George C. · 1 · May 28 2015
• What is a sequence?

  A sequence is an infinite set of points which are ordered and a mapping can be associated with with that set and the set of natural numbers.

  The general notion of a sequence is that it is an infinite set with every element associated with a natural number even though all infinite sets may not be sequences.

  A sequence may be represented by, `{x_n}` where `x_n` is the `n`th element related to the a corresponding natural number.

  Thus if `x_n = 1/n^2`, the sequence may be given as,

  `{1,1/4,1/9,1/16,....}`

  For `x_n = n^3` we shall have,

  `{1, 8, 27, 64,....}`

  Now for `x_n = n` we can have,

  `{1,2,3,.....}`

  This is indeed the set of naturals.

  However, there can be other ordered arrays of numbers which are sometimes referred at as sequences. They don't fit right with the definition I gave.

  Let's for example take a Fibonacci sequence.

  `1,1,2,3,5,8,13,....`

  This sequence is made by adding the previous two numbers on the list to form the next one and so on.

  There can be arithmetic sequences, like

  `2,8,14,20,....` which has first term `2` and common difference `6`.

  I like to call them progressions and reserve the word sequence for the definition I proposed in the first 2 paragraphs.

  Aritra G. · 20 · Sep 7 2017
• How do you find the general term for a sequence?

  Answer:

  It depends.

  Explanation:

  There are many types of sequences. Some of the interesting ones can be found at the online encyclopedia of integer sequences at https://oeis.org/

  Let's look at some simple types:

  Arithmetic Sequences

  `a_n = a_0 + dn`

  e.g. `2, 4, 6, 8,...`

  There is a common difference between each pair of terms.

  If you find a common difference between each pair of terms, then you can determine `a_0` and `d`, then use the general formula for arithmetic sequences.

  Geometric Sequences

  `a_n = a_0 * r^n`

  e.g. `2, 4, 8, 16,...`

  There is a common ratio between each pair of terms.

  If you find a common ratio between pairs of terms, then you have a geometric sequence and you should be able to determine `a_0` and `r` so that you can use the general formula for terms of a geometric sequence.

  Iterative Sequences

  After the initial term or two, the following terms are defined in terms of the preceding ones.

  e.g. Fibonacci

  `a_0 = 0`
  `a_1 = 1`
  `a_(n+2) = a_n + a_(n+1)`

  For this sequence we find: `a_n = (phi^n - (-phi)^(-n))/sqrt(5)` where `phi = (1+sqrt(5))/2`

  There are many ways to make these iterative rules, so there is no universal method to provide an expression for `a_n`

  Polynomial Sequences

  If the terms of a sequence are given by a polynomial, then given the first few terms of the sequence you can find the polynomial.

  e.g.

  `color(red)(1), 2, 4, 7, 11,...`

  Form the sequence of differences of these values:

  `color(red)(1), 2, 3, 4,...`

  Form the sequence of differences of these values:

  `color(red)(1), 1, 1,...`

  Once you reach a constant sequence like this, pick out the initial terms from each sequence. In this case `1`, `1` and `1`.

  These form the coefficients of a polynomial expression:

  `a_n = color(red)(1)/(0!) + (color(red)(1)*n)/(1!) + (color(red)(1)*n(n-1))/(2!)`

  `=n^2/2+n/2+1`

  George C. · 3 · Jul 18 2015

• What is a sequence?
• How does the Fibonacci sequence relate to Pascal's triangle?
• What is the Fibonacci sequence?
• How do I find the `n`th term of the Fibonacci sequence?
• How do you find the general term for a sequence?
• How do find the `n`th term in a sequence?
• What is the golden ratio?
• How does the golden ratio relate to the Fibonacci sequence?
• How do you determine if -10,20,-40,80 is an arithmetic or geometric sequence?
• How do you determine if 15,-5,-25,-45 is an arithmetic or geometric sequence?
• A man went to his bank and deposited $1,663. A couple of days later, that same man deposited $1,527. A few days after that, he deposited $1,126. A couple days later, he deposited $1,096. What was his next deposit?
• What is the next number in the sequence 72, 36, 18, ____, 4.5?
• What is the next number in the sequence 625, 25, ____?
• What is the next number in the sequence ___, ____, 16,256, 65,536?
• What is the next number in the sequence ___, ___, 16, 25, 36?
• What is the next number in this sequence 1, 1, 3, 2, 4, 6, 5, 25?
• What is the pattern in the sequence 1, 3, 4, 7, 6, 12, 8, 15, 13, ??
• What is the pattern in the sequence 100, 19, 83, 34, 70, 45?
• What is the pattern in the sequence 1, 1, 0, -1, 0, 7, 28, 79, 192?
• What is the pattern in the sequence 2, 5, 10, 17, 28, 41, 58, 77, 100?
• What is the pattern in the sequence 1 2 4 3 6 8 7 14 16?
• How do you find the sum of the first 25 terms of an arithmetic sequence whose 7th term is −247 and whose 18th term is −49?
• For the sequence an = 4n + 5 how do you find the first five terms?
• What is the formula for the sequence `2, -1, 4, -7, 10, -13, 16,...` ?
• The sequence `0, 2, 8, 30, 112, 418,...` is defined recursively by `a_0 = 0`, `a_1 = 2`, `a_(n+1) = 4a_n-a_(n-1)`. What is the formula for a general term `a_n`?
• How do you write a recursive formula for a sequence whose first three terms are 1, 1, and 3?
• How do you find the first five terms of each sequence `a_1=12`, `a_(n+1)=a_n-3`?
• How do you find the first five terms of each sequence `a_1=-3`, `a_(n+1)=a_n+n`?
• How do you find the first five terms of each sequence `a_1=0`, `a_(n+1)=-2a_n-4`?
• How do you find the first five terms of each sequence `a_1=1`, `a_2=2` `a_(n+2)=4(a_(n+1))-3a_n`?
• How do you find the first three iterates for the function `f(x)=3x-4` given the initial value `x_0=3`?
• How do you find the first three iterates for the function `f(x)=-2x+5` given the initial value `x_0=2`?
• How do you find the first three iterates for the function `f(x)=x^2+2` given the initial value `x_0=-1`?
• How do you find the first five terms given `a_1=-6` and `a_(n+1)=a_n+3`?
• How do you find the first five terms given `a_1=13` and `a_(n+1)=a_n+5`?
• How do you find the first five terms given `a_1=2` and `a_(n+1)=a_n-n`?
• How do you find the first five terms given `a_1=6` and `a_(n+1)=a_n+n+3`?
• How do you find the first five terms given `a_1=9` and `a_(n+1)=2a_n-4`?
• How do you find the first five terms given `a_1=4` and `a_(n+1)=3a_n-6`?
• How do you find the first five terms given `a_1=-1`, `a_2=5`, `a_(n+1)=a_n+a_(n-1)`?
• How do you find the first five terms given `a_1=4`, `a_2=-3`, `a_(n+2)=a_(n+1)+2a_n`?
• How do you find the first five terms given `a_1=3/4`, `a_(n+1)=(n^2+1)/n*a_n`?
• If `a_0=7` and `a_(n+1)=a_n+12` for `n>=0`, how do you find the values of `a_5`?
• How do you find the first three iterate of the function `f(x)=9x-2` for the given initial value `x_0=2`?
• How do you find the first three iterate of the function `f(x)=4x-3` for the given initial value `x_0=2`?
• How do you find the first three iterate of the function `f(x)=3x+5` for the given initial value `x_0=-4`?
• How do you find the first three iterate of the function `f(x)=2x^2-5` for the given initial value `x_0=-1`?
• How do you find the first three iterate of the function `f(x)=3x^2-4` for the given initial value `x_0=1`?
• How do you find the first three iterate of the function `f(x)=3x^2-3x+2` for the given initial value `x_0=1/3`?
• If the rate of inflation is 2%, the cost of an item in future years can be found by iterating the function `c(x)=1.02x`, how do you find the cost of a $70 portable stereo in four years if the rate of inflation remains constant?
• How do you find the first five terms of the sequence `a_1=7`, `a_(n+1)=a_n-2`?
• How do you find the first five terms of the sequence `a_1=3`, `a_(n+1)=2a_n-1`?
• Apart from `2, 3` and `3, 5` is there any pair of consecutive Fibonacci numbers which are both prime?
• How do you express the sequence below as a recursively defined function 4, 11, 25, 53, 109,...?
• How do you find the nth term of the sequence `-1,2,-3,4,-5,6,...`?
• The nth term of a sequence is `2^n+2^(n-1)`, how do you work out the 10th term of the sequence?
• How do you identify `a_3` of this sequence: 0.25, 0.5, 0.75, 1, 1.25, 1.5,?
• How do you determine whether each sequence could be geometric, arithmetic, or neither. Use a patte to write the next three terms. 460, 46, 4.6, 0.46?
• What is the missing sequence in `1 3/8, 1 3/4, 2 1/8, __, 2 7/8`?
• Which of the following is the algebraic expression that best describes the sequence 7,14, 21, 28,?
• How do you write the first five terms of the sequence `a_n=3n+1`?
• How do you write the first five terms of the sequence `a_n=5n-3`?
• How do you write the first five terms of the sequence `a_n=2^n`?
• How do you write the first five terms of the sequence `a_n=(1/2)^n`?
• How do you write the first five terms of the sequence `a_n=(-1/2)^n`?
• How do you write the first five terms of the sequence `a_n=(n+2)/n`?
• How do you write the first five terms of the sequence `a_n=n/(n+2)`?
• How do you write the first five terms of the sequence `a_n=(6n)/(3n^2-1)`?
• How do you write the first five terms of the sequence `a_n=(3n^2-n+4)/(2n^2+1)`?
• How do you write the first five terms of the sequence `a_n=(1+(-1)^n)/n`?
• How do you write the first five terms of the sequence `a_n=1+(-1)^n`?
• How do you write the first five terms of the sequence `a_n=2-1/3^n`?
• How do you write the first five terms of the sequence `a_n=2^n/3^n`?
• How do you write the first five terms of the sequence `a_n=1/(n^(3/2))`?
• How do you write the first five terms of the sequence `a_n=10/n^(2/3)`?
• How do you write the first five terms of the sequence `a_n=3^n/(n!)`?
• How do you write the first five terms of the sequence `a_n=(n!)/n`?
• How do you write the first five terms of the sequence `a_n=(-1)^n(n/(n+1))`?
• How do you write the first five terms of the sequence `a_n=2/3`?
• How do you write the first five terms of the sequence `a_n=0.3`?
• How do you write the first five terms of the sequence `a_n=n(n-1)(n-2)`?
• How do you write the first five terms of the sequence `a_n=n(n^2-6)`?
• How do you find `a_25` given `a_n=(-1)^n(3n-2)`?
• How do you find `a_16` given `a_n=(-1)^(n-1)[n(n-1)]`?
• How do you find `a_10` given `a_n=2^n/(n!)`?
• How do you find `a_8` given `a_n=(n!)/(2n)`?
• How do you find `a_11` given `a_n=(4n)/(2n^2-3)`?
• How do you find `a_13` given `a_n=(4n^2-n+3)/(n(n-1)(n+2))`?
• How do you write the expression for the nth term of the sequence given `1, 4, 7, 10, 13,...`?
• How do you write the expression for the nth term of the sequence given `3, 7, 11, 15, 19,...`?
• How do you write the expression for the nth term of the sequence given `0, 3, 8, 15, 24,...`?
• How do you write the expression for the nth term of the sequence given `2, -4, 6, -8, 10,...`?
• How do you write the expression for the nth term of the sequence given `-2/3, 3/4, -4/5, 5/6, -6/7,...`?
• How do you write the expression for the nth term of the sequence given `2/1, 3/3, 4/5, 5/7, 6/9,...`?
• How do you write the expression for the nth term of the sequence given `1/3, 2/9, 4/27, 8/81,...`?
• How do you write the expression for the nth term of the sequence given `1, 1/4, 1/9, 1/16, 1/25,...`?
• How do you write the expression for the nth term of the sequence given `1, 2, 2^2/2, 2^3/6, 2^4/24, 2^5/120,...`?
• How do you write the expression for the nth term of the sequence given `1+1/2, 1+3/4, 1+7/8, 1+15/16, 1+31/32,...`?
• How do you write the first five terms of the sequence defined recursively `a_1=28, a_(k+1)=a_k-4`?
• How do you write the first five terms of the sequence defined recursively `a_1=15, a_(k+1)=a_k+3`?
• How do you write the first five terms of the sequence defined recursively `a_1=3, a_(k+1)=2(a_k-1)`?
• How do you write the first five terms of the sequence defined recursively `a_1=32, a_(k+1)=1/2a_k`?
• How do you write the first five terms of the sequence defined recursively `a_1=6, a_(k+1)=a_k+2`, then how do you write the nth term of the sequence as a function of n?
• How do you write the first five terms of the sequence defined recursively `a_1=25, a_(k+1)=a_k-5`, then how do you write the nth term of the sequence as a function of n?
• How do you write the first five terms of the sequence defined recursively `a_1=81, a_(k+1)=1/3a_k`, then how do you write the nth term of the sequence as a function of n?
• How do you write the first five terms of the sequence defined recursively `a_1=14, a_(k+1)=(-2)a_k`, then how do you write the nth term of the sequence as a function of n?
• Assume that `f_1,f_2,...,f_n,...` are the Fibonacci numbers. Then `AAn in NN, f_(4n)` is a multiple of `3`. Prove this statement is true?
• Sum of `n` terms of a certain series is given by `S_n=2n+3n^2`, what is the type of the series and what is its `20^(th)` term?
• How do I find the greatest lower bound for the sequence `A={\frac{1}{n+10}}_{n=1}^{\infty}`?
• `4`, `29`, `129`, `354`, `754`....?