albees046

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Number sequences, monotonicity and boundedness


A number sequence (an) is called monotonically increasing or monotonically decreasing if for all n∈N:


an + 1≥an or an + 1≤an, respectively.


If each sequence member is genuinely larger (smaller) than its predecessor - do your homework , the sequence is said to be strictly monotonically increasing (decreasing). A sequence of numbers (an) is called upper bounded or lower bounded if and only if there exists a number s∈R such that for all sequence members an holds:


an≤s resp. an≥s


The real number s is then called an upper or a lower bound of the number sequence - excel homework help . We consider the sequences of numbers that result from the processes or situations shown below:


(1) A basic fare of 1.50 euros and an additional 0.60 euros for each kilometer must be paid for a trip by cab in A-city.


The fare for a trip of 1 km, 2 km, 3 km, ..., 10 km is therefore 2.10 Euro, 2.70 Euro, 3.30 Euro, ..., 7.50 Euro.


(2) A bookstore has 200 copies of a certain book in stock. In the course of a week, 34 books are sold on the first day, 25 books on the second day, 11 books on each of the third and fourth days, no books on the fifth day and 4 books on the sixth day.


The stock of books during the week is:


200; 166; 141; 130; 119; 119; 115


(3) Students take a measurement of air temperature on two days at 9 am; 12 pm, 3 pm and 6 pm respectively. They obtain the following values (in ° C): 7; 18; 19; 12; 9; 19; 20; 17 (Fig. 3).


Comparing the numerical values in these three examples, we can see that in (1) the links of the price sequence are constantly increasing, in (2) the inventory decreases from day to day or at least remains the same, but in (3) no such regularity occurs.


This leads to the following conceptualizations (definitions):


A sequence of numbers (an) is called monotonically increasing if and only if for all n∈N :
 an + 1≥an or an + 1-an≥0.


A sequence of numbers (an) is called monotonically decreasing if for all n∈N holds:
 an + 1≤an or an + 1-an≤0, respectively.


For monotonically increasing or monotonically decreasing sequences - homework help geometry , successive sequence members may be equal. If each sequence member is genuinely larger (smaller) than its predecessor, we speak of strictly monotonically growing (decreasing) sequences.


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