Essential Mathematics: Sequences And Series

24 flashcards · Mathematics · by lakesare

[Ways to specify a sequence] Closed form VS recurrence system?
closed form - defines the general term an in terms of n. an = 2n − n2 (n = 1, 2, 3)To specify a sequence using a closed form, two pieces of information are needed: 1. the closed form; 2. the range of values for the subscript n.recurrence system - defines an (other than the first) in terms of the previous term. a1 = 1, an = 5an−1 − 2 (n = 2, 3, 4)To specify a sequence using a recurrence system, three pieces of information are needed: 1. the value of the first term a1 2. the recurrence relation 3…
[Arithmetic sequence] Name 2 parameters of the arithmetic sequence.
Choosing the values of: first term x1, and common difference d = xn - xn-1 determines a particular arithmetic sequence.
[Geometric sequence] Name 2 parameters of the geometric sequence.
Choosing the values of: first term x1, and common ratio r = xn / xn-1 determines a particular geometric sequence.
[Arithmetic sequence] Name 2 good analogies for the arithmetic sequence. 7.15, 10.50, 13.85, ~ 1000, 950, 900, ~
This sequences could represent: heights of successive floors in a very tall building, from the first floor upwards volume of oil in a tank supplying a boiler that uses 50 litres of oil per week.
[Arithmetic sequence] For the infinite arithmetic sequence (xn) whose first four terms are 100, 95, 90, 85, write down the corresponding recurrence system.
To specify a sequence using a recurrence system, 3 pieces of information are needed: the value of the first term a1 the recurrence relation the range of values for the subscript n.
[Geometric sequence] Name 2 good analogies for the geometric sequence. 1000, 1050, 1102.5, 1157.625, ~ 2, 1.4, 0.98, 0.686, 0.4802
This sequences could represent: amount of money (in £) in a savings account - amount added each year is 0.05 times (that is, 5% of) the amount last year. heights, measured in meters, of successive bounces of a ball that is assumed to rebound to 70% of the height from which it falls.
[Arithmetic sequence] Name long-term behaviours of arithmetic sequences?
[Geometric sequence] Name long-term behaviours of geometric sequences?
What's series?
A series is the addition of consecutive terms of a sequence. The singular and plural forms of the word 'series' are the same.The number that you obtain when you add up all the terms of a series is called the sum of the series, and the process of finding this sum is called summing the series, or evaluating the series.
Sequence VS series?
sequence - a list of numbers (e.g. 1000, 950, 900) series - adding the numbers from the sequence (e.g. 1000 + 950 + 900) Series are usually represented using the summation sign. The number that you obtain when you add up all the terms of a series is called the sum of the series, and the process of finding this sum is called summing the series, or evaluating the series.
[Arithmetic sequence] What's the intuition behind this formula of arithmetic series?
Reverse the order of the terms of the series, and write the result under the original series, with the terms aligned.The reason for all the 49s is that: each term in the original series is 3 more than the one before, whereas each term in the reverse series is 3 less than the one before.
[Geometric sequence] What's the intuition behind this formula of geometric series?
Multiply the series by the common ratio r, and write the result under the original series, with terms aligned. The subtract s - rs.
What's a geometric series?
A geometric series is a series whose terms come from a geometric sequence.
[Geometric sequence] Which of these infinite geometric series have a sum?
c) and d)
[Arithmetic sequence, geometric sequence] What's the sum of infinite arithmetic series VS infinite geometric series? The number that you obtain when you add up all the terms of a series is called the sum of the series, and the process of finding this sum is called summing the series, or evaluating the series. For example, the sum of the series 1 + 3 + 5 + 7 is 16. You may think that this terminology is a little strange, because a series is already a sum, but it’s standard terminology, and is co…
arithmetic - arithmetic series only have a sum when the first term is 0, and common difference is 0 (that sum is 0). a + (a + d) + (a + 2d) + ··· The situation for infinite arithmetic series is much simpler than for infinite geometric series. In summary, the only infinite arithmetic series that has a sum is the infinite series with first term 0 and common difference 0, that is, the series 0 + 0 + 0 + ··· , which has sum 0.geometric - geometric series only have a sum when common ratio is r ∈ (0,…
[Geometric sequence] How would you go about deciding whether this infinite geometric series has a sum (and what sum)?
Use the formula for the sum of a finite geometric series with n = ∞.
What operations are used for recurrence system VS closed form notation for: Arithmetic sequence Geometric sequence?
Arithmetic sequence Recurrence system - addition + d Closed form - multiplication d * n Geometric sequence Recurrence system - multiplication * r Closed form - power rn
What are triangular numbers?
Triangular number count objects arranged in an equilateral triangle. Thus triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The first six triangular numbers.
Monomial VS binomial VS polynomial?
Polynomials can be classified by the number of terms with nonzero coefficients: monomial axm binomial axm + bxn polynomial axm + bxn + ... + ~x~ Polynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a monomial, a two-term polynomial is called a binomial, and a three-term polynomial is called a trinomial.
[Binomial theorem] What's the binomial theorem?
The binomial theorem describes the expansion of powers of a binomial. A binomial is an expression that is the sum of two terms, so (a + b)n is a power of a binomial.
[Binomial theorem] What does the Pascal's triangle denote?
Coefficients in the binomial expansion of various n-s. The right-hand side of the equation in the binomial theorem is known as the binomial expansion of (a + b)n.
[Binomial theorem] Name 2 interesting facts about the Pascal's triangle.
Powers in each term sum up to n. Each new formula is produced by multiplying the formula before by a + b. When this is done, each term of the old formula is multiplied by a to give terms of the new formula, and separately multiplied by b to give further terms of the new formula (before the like terms are collected). This raises the sum of the powers in each term by 1.Each coefficient in the new formula is a sum of two coefficients higher up: nCk = n-1Ck-1 + n-1Ck. You can see that each coeffici…
[Binomial theorem] What are binomial coefficients?
Coefficients of the terms in binomial expansion. Pascal's triangle. In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem.
[Finding a particular term in another binomial expansion] How would you go about finding the coefficient of p4 in the expansion of this power of a binomial?
By expressing a generic term in the binomial expansion with k.