What Number Fills in the Blank to Complete the Factorization of 3x + 24? (X + 8)
Number Sequence Calculator
Arithmetic Sequence Calculator
definition: an = ai + f × (north-i)
example: 1, 3, 5, 7, ix 11, thirteen, ...
Geometric Sequence Reckoner
definition: anorth = a × rnorthward-i
example: 1, ii, iv, eight, 16, 32, 64, 128, ...
Fibonacci Sequence Calculator
definition: a0=0; a1=1; an = an-i + anorthward-two;
example: 0, 1, ane, 2, three, 5, 8, thirteen, 21, 34, 55, ...
In mathematics, a sequence is an ordered list of objects. Accordingly, a number sequence is an ordered list of numbers that follow a item blueprint. The individual elements in a sequence is ofttimes referred to as term, and the number of terms in a sequence is called its length, which can be infinite. In a number sequence, the guild of the sequence is important, and depending on the sequence, information technology is possible for the same terms to appear multiple times. There are many different types of number sequences, three of the about common of which include arithmetics sequences, geometric sequences, and Fibonacci sequences.
Sequences have many applications in various mathematical disciplines due to their backdrop of convergence. A serial is convergent if the sequence converges to some limit, while a sequence that does not converge is divergent. Sequences are used to study functions, spaces, and other mathematical structures. They are peculiarly useful as a basis for series (essentially depict an functioning of calculation infinite quantities to a starting quantity), which are generally used in differential equations and the area of mathematics referred to as analysis. There are multiple ways to denote sequences, 1 of which involves simply listing the sequence in cases where the pattern of the sequence is easily discernible. In cases that accept more than complex patterns, indexing is usually the preferred note. Indexing involves writing a full general formula that allows the decision of the nthursday term of a sequence as a function of n.
Arithmetic Sequence
An arithmetic sequence is a number sequence in which the difference between each successive term remains constant. This difference can either be positive or negative, and dependent on the sign will result in terms of the arithmetic sequence disposed towards positive or negative infinity. The general form of an arithmetic sequence tin can exist written equally:
| an = a1 + f × (n-one) or more more often than not | where an refers to the northth term in the sequence | |
| an = am + f × (northward-thousand) | aone is the get-go term | |
| i.east. | a1, aone + f, ai + 2f, ... | f is the common difference |
| EX: | 1, 3, five, 7, 9, eleven, 13, ... | |
It is clear in the sequence in a higher place that the mutual difference f, is 2. Using the equation above to calculate the fiveth term:
| EX: | afive = a1 + f × (due north-1) a5 = one + two × (5-one) afive = one + 8 = 9 | |
Looking dorsum at the listed sequence, it can be seen that the 5th term, av , found using the equation, matches the listed sequence as expected. It is also ordinarily desirable, and uncomplicated, to compute the sum of an arithmetic sequence using the following formula in combination with the previous formula to notice adue north :
Using the aforementioned number sequence in the previous example, find the sum of the arithmetic sequence through the 5th term:
| EX: | 1 + 3 + 5 + 7 + 9 = 25 (five × (one + 9))/2 = 50/2 = 25 | |
Geometric Sequence
A geometric sequence is a number sequence in which each successive number after the first number is the multiplication of the previous number with a stock-still, non-zero number (mutual ratio). The general form of a geometric sequence tin can be written equally:
| an = a × rn-i | where anorth refers to the due northth term in the sequence | |
| i.e. | a, ar, artwo, ar3, ... | a is the calibration gene and r is the common ratio |
| EX: | ane, 2, four, eight, 16, 32, 64, 128, ... | |
In the case above, the mutual ratio r is 2, and the scale factor a is one. Using the equation above, summate the 8th term:
| EX: | aeight = a × r8-1 a8 = i × 27 = 128 | |
Comparison the value found using the equation to the geometric sequence above confirms that they match. The equation for calculating the sum of a geometric sequence:
Using the same geometric sequence in a higher place, notice the sum of the geometric sequence through the threerd term.
EX: one + 2 + 4 = seven
Fibonacci Sequence
A Fibonacci sequence is a sequence in which every number post-obit the first two is the sum of the two preceding numbers. The commencement 2 numbers in a Fibonacci sequence are defined as either ane and i, or 0 and 1 depending on the chosen starting point. Fibonacci numbers occur oftentimes, too as unexpectedly within mathematics and are the subject of many studies. They have applications within computer algorithms (such as Euclid's algorithm to compute the greatest common factor), economics, and biological settings including the branching in trees, the flowering of an artichoke, also every bit many others. Mathematically, the Fibonacci sequence is written as:
| an = anorthward-1 + an-two | where anorth refers to the nth term in the sequence | |
| EX: | 0, 1, 1, 2, 3, 5, 8, 13, 21, ... | a0 = 0; a1 = i |
Source: https://www.calculator.net/number-sequence-calculator.html
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