Some visual proofs of Pythagoras' Theorem My favourite proof of the look-and-see variety is on the right.
The recursive formula is the mathematical equation for the verbal description. Both the verbal description and the recursive formula will uniquely determine the values in the list. However, different people may observe different patterns in a list, and come to different conclusions about the verbal description or the recursive formula used to represent that list.
The list can be written as a sequence of ordered pairs, where the first coordinate denotes the position in the sequence, and the second coordinate is the value of the number at that position. We can then see that a sequence is a set of points that can be graphed, and this suggests that an equation can be written that describes the sequence value as a function of its position in the list.
The explicit formula is that function. The Definition of a Sequence Having considered various descriptions of a sequence, we are now ready to provide some official definitions. A sequence is a function whose domain is the set of natural numbers.
A term of the sequence is a specific value in the sequence. The natural numbers are the counting numbers, namely 1, 2, 3, 4, and so on. In other words, a sequence has a restricted domain. The list begins with a first term, is followed by a second term, and a third term, and so on.
It is not possible to have a term in a fractional position, or in a negative position. Sometimes, people will introduce other positions, and it is rather common to see some sequences defined to have a zeroth position. Nevertheless, the idea of a restricted domain is still present, and is a hallmark of a sequence.
Recursive and Explicit Formulas Two different types of formulas are commonly used to describe sequences. The recursive formula describes each term of the sequence in relation to the terms immediately preceding it.
It is not necessary to know the position in the list to obtain the next value. On the other hand, the explicit formula does depend on the position of the list.
It is not necessary to know the previous term in the list. But if we had the previous term of the list and did not know its position, we would have to first use the formula to determine that position before we could find the next term.
In other words, recursive formulas and explicit formulas serve two very different purposes. Standard Types of Problems Most problems involving sequences involve one or more of the following types: Finding specific terms of the sequence.
When a formula is known, appropriate substitutions will solve this problem. Finding a formula for the sequence. This situation can be quite a challenge, and a successful solution may involve a fair amount of intuition.
However, there are some techniques that can help.
Finding the eventual behavior of a sequence. If an explicit formula is known, then the behavior of the function as the variable approaches infinity can be examined. In general, the sequence can be unbounded, attracted to a fixed point,cycle between two or more values, or fluctuate in a chaotic pattern.
See the web page Recursion and Cobweb Diagrams for more information. Adding the terms of a sequence. Generally, if this problem begins with a list of numbers, then the formula will need to be obtained, and this can be a challenge.Two ways you can solve this problem are visually and mathematically.
Both will give you the final answer of Visual Method Between each of the numbers, the difference is increasing by 2 each time.
3 comes 2 after 1, 7 comes 4 after 3, 13 comes 6 after 7, and so on. Since 43 comes 12 after 31, the next number would come 14 after Because of this, 43 + 14 is 57 –– your answer. Status. This is a work in progress release of the GnuCOBOL FAQ.
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GnuCOBOL is the release version. The recursive formula for the given sequence above which is 7, 13, 19, 25, and 31 and the next term in the sequence is this: an = an – 1 + 6, where a1 = 7; 37 Here is the solution: f(1) = /5(2).
While recursive sequences are easy to understand, they are difficult to deal with, in that, in order to get, say, the thirty-nineth term in this sequence, you would first have to find terms one through thirty-eight.
There isn't a formula into which you could plug n . Ans: e. 2 + 5 + 8 + 11 + + 41 Ans: f.
7 + 1 5 11 17 23 Ans: Show that each sequence is geometric, then nd the common ratio r and. This test includes recursive and general formula, generating the sequence, finding the nth term, arithmetic and geometric sequence and application.