Series
Arithmetic Mean Definition:
Arithmetic mean is commonly called as average. Mean or Average is defined as the sum of all the given elements divided by the total number of elements.
Formula:
Mean = sum of elements / number of elements
= a1+a2+a3+.....+an/n
Example: To find the mean of 3,5,7
15/3 = 5
Note:
- Mean is defined only for real positive number input. No negative numbers is allowed as input
- The number of input is finite up to n. (not infinite)
- The mean value is somewhere in between the lowest and the highest input numbers. The mean value will not go outside this range of input.
Geometric Mean
The geometric mean is the nth root of the product of the scores. Thus, the geometric mean of the scores: 1, 2, 3, and 10 is the
fourth root of 1 x 2 x 3 x 10 which is the fourth root of 60 which equals 2.78. The formula can be written as:
GM= (a1*a2*a3*.......an)1/n The geometric mean between two numbers is: √(f1 · f2)
Harmonic mean Harmonic mean is the reciprocal of arithmetic mean of reciprocal.
NOTE: 1. If the data is all positive, the arithmetic mean is always greater than or equal to the geometric mean of the same data
2. The log of the geometric mean of a set of numbers is the arithmetic mean of the logs of the numbers:
log((abc)^(1/3)) = (1/3)log(abc) = (1/3)(log(a)+log(b)+log(c))
Arithmetic Series
A military unit purchases 10 spare parts during the first month of a contract, 15 spare parts in the second month, 20 spare parts in the third month, 25 spare parts in the fourth month, and so on. The acquisition officer wants to know the total number of spare parts the unit will have acquired after 50 months. This sequence of number of parts purchased in each month is called an Arithmetic Series and the sum of this series (i.e., the total number of purchased spare parts) can be written as follows.
S = 10 + [10+(1*5)] + [10+(2*5)] + [10+(3*5)] + ... +[10+(49*5)]
and can be simplified as:
Tn= [i + (n-1)*d]
sn = n(t1 + tn)/2
-
i is the initial term,
-
Tn is the nth term of the series
d is the difference between successive terms and
n is the number of terms in the series.
where
Applying this sum formula to the above example, we get
Geometric Series
Sequences of numbers that follow a pattern of multiplying a fixed number from one term to the next are called geometric sequences. The following sequences are geometric sequences:Sequence A: 1 , 2 , 4 , 8 , 16 , ...
Sequence B: 0.01 , 0.06 , 0.36 , 2.16 , 12.96 , ...
For sequence A, if we multiply by 2 to the first number we will get the second number. This works for any pair of consecutive numbers. The second number times 2 is the third number: 2 × 2 = 4, and so on.
Because these sequences behave according to this simple rule of multiplying a constant number to one term to get to another, they are called geometric sequences. So that we can examine these sequences to greater depth, we must know that the fixed numbers that bind each sequence together are called the common ratios. Mathematicians use the letter r when referring to these types of sequences. Mathematicians also refer to generic sequences using the letter a along with subscripts that correspond to the term numbers as follows:
The formula for the general term for each geometric sequence is An=A1*Rn-1
The Sum of a Geometric Series Sn= [A1(1-Rn)] / (1-R)
= A1 / (1-R) For an infinity series ie when n is infinity
Generic Sequence: a1, a2, a3, a4, ...
This means that if we refer to the tenth term of a certain sequence, we will label it a10. a14 is the 14th term. This notation is necessary for calculating nth terms, or an, of sequences. r can be calculated by dividing any two consecutive terms in a geometric sequence. The formula for calculating r is . . R=(An/An-1)which geometric series is
which geometric series is having 1 as comman ratio????