Today we are going to learn about how the sequence of various numbers work and how we can trace out what Number would follow after a certain number , or at a particular index,
PROGRESSION
Sequence following certain patterns are called Progression.
e.g. 2,3,4,5,..... is a progression, here each term is interesting, here each term is increasing by 1.
ARITHMETICAL PROGRESSION
An arithmetic progression is a sequence in which the difference between any term and its just preceding term is constant through. The constant 'd' is called the common difference. The first term of AP is represented by 'a' and the formula for the xth term is
aₓ = a + (x - 1) d
If an AP has first term = a , and common difference = d then the general form of an AP is
a , a + d , a + 2d , a + 3d , ......... , a + (x - 1)d
* Sum of the n terms of an AP is ,
= (x / 2) * (2a + (x - 1)d)
* If first term = a and last term = aₓ are given , then sum of x terms of an AP is
Sₓ = x ( a₁ + aₓ ) / 2
where x = number of terms
The n th term of a series is equal to the sum of n terms subtract the sum of (x - 1) terms i.e.
aₓ = Sₓ - Sₓ₋₁
ARITHMETIC MEAN (AM)
When terms are in AP , then the middle term is called Arithmetic mean.
So, if a,b,c are in AP . Then AM,
b = (a + c )/ 2 => 2b = a + c
'b' is AM between a and c.
GEOMETRIC SEQUENCE
A geometric sequence is a sequence of numbers, whose first term is non zero and each of the term is obtained by multiplying its just preceding its just preceding term by a constant quantity.
This constant quality. This constant quatity is called the common ratio of the GP. Thus , if t1,t2,t3 _ are in GP then common ratio ,
r = second term / first term
If a is the first term and r is the common ratio then GP can be written as a , ar , ar², ar³.... arⁿ⁻¹
(*) nth term of an GP is Tn =arⁿ⁻¹ = L
(*) Sum of the n terms of an GP is Sn = a(rⁿ - 1) / (r -1)
(*) Sum of infinite terms of an GP is Sn = a / (1 - r)
(*) nth term of a GP from end = L / rⁿ⁻¹
GEOMETRIC MEAN
If three terms are in GP then the middle term is called the geometric mean. If a, b, c are in GP then b is the GM of a and c.
Let a and b be two numbers and G be the GM between them.
Then a , G , b are in GP,
G = √ab , a > 0 , b > 0
HARMONIC PROGRESSION (HP)
A sequence is said to be harmonic progression (HP) . If the reciprocals of its terms are in Arithmetic Progression (AP) e.g. The sequence 1, 1/3, 1/5, 1/7, .... is an HP because the sequence 1 , 3 , 5 , 7 ... is an AP .
If a1, a2, a3 ,.... ,an are in HP then , 1/a1, 1/a2, 1/a3, .... 1/an are in AP .
(*) nth term of an HP is Tn = 1 / {(1/a1) - (n - 1)(1/a2 - 1/a1 ) }
HARMONIC MEAN
If three terms are in HM then the middle term is called the harmonic mean if a, b ,and c are in HM then b is the HM of a and c.
Let a and b be two number H be the HM between them.
Then a, b are in HP.
H = 2ab / (a + b)
Relation Between Arithmetic, Geometric and Harmonic Mean
Let A, G and H be the arithmetic, geometric and harmonic means between a and b , then
* A ≥ G ≥ H
* G² = AH
SUM TO N TERMS OF SPECIAL SERIES
The sum of first n terms of special series is given below,
* sum of first n natural numbers = 1 + 2 + 3 + 4 + .... + n = n(n + 1)/ 2
* sum of square of the first n natural numbers = 1² + 2² + 3² + ..... + n² = n (n + 1) (2n + 1) / 6
,
* sum of cubes of the first n natural numbers = 1³ + 2³ + 3³ + ..... + n³ = [n(n + 1)/ 2]²
Thanks for reading this at Math Capsule
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