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AP Series , GP Series and HP Series Formula

Definition of Arithmetic Progressions (A.P) Series

A series of number is termed to be in arithmetic progression when the difference between two consecutive numbers remain the same.

It also means that the next number can be obtained by adding or subtracting the constant number to the previous in the sequence. Therefore, this constant number is known as the common difference(d). 

For example, 3, 6, 9, 12 is an AP as the difference between two consecutive terms is three which is fixed. 

• Read Also  – Tips & tricks to solve AP GP & HP questions 

Arithmetic Progressions Series Formula (AP Series formula )

Suppose if, ‘a’ is the first term and ‘d’ is a common difference, then,

• Formula to find Nth term of arithmetic progression is = a + (n -1) d
• Arithmetic mean = (Sum of every term in the series of AP /  Total number of terms in the AP)
• Formula to find Sum of ‘n’ terms of an AP = [nx (first term + last term)]/2

• Formula to find the sum of given terms in an AP = s=[n(a+L)/2]

Where, L = a + (n – 1) d Thus,

s=[nx{2a+(n-1)d}/2]

• Formula to find Number of terms of an AP n ={(l – a)/d} +  1  where l : last term , a : first term and d : common difference

• T_n = S_n – S_n-1 , where T_n = nth term: t_n= a + (n – 1)d
• If a, b and c are three terms in AP then b = (a+c)/2 which is called Arithmetic Mean

Definition of Geometric Progression (G.P) Series

If the ratio of any two successive terms is invariably similar, then the sequence is termed a geometric progression.

Naturally, it can be defined as a series in which the next number in the series can be obtained by multiplying a constant to the previous number in the series. Therefore, this fixed term is known as the common ratio. Quantities are termed as GP if they tend to increase and decrease by a common factor.

For example, 2, 4, 8, 16 can be a GP because the ratio between two consecutive numbers is same, i.e. 2.

Geometric Progression (G.P) Series Formula

Suppose, if ‘a’ is the first term and ‘r’ be the common ration, then

• Formula for nth term of GP = a r ^n-1

Geometric mean = nth root of the product of ‘n’ terms in the GP.

• Formula to find the geometric mean between two quantities

Let the two quantities be ‘a’ and ‘b’. Then if a, G, and b are in GP

b/G = G/a

Each of the ones will be equal to the common ration,

G² = ab

Therefore, G = √ab 

• Formula to find the sum of the number of terms in a GP

Let ‘a’ be the first term, ‘r’ be the common ration and ‘n’ be the number of terms

1. If r > 1 then,
       S_{n =}a(rⁿ-1)/(r -1)
2.  If r < 1 , then 
       S_{n =}a(1 – rⁿ)  / (1 -r)
3. Sum of infinite terms in a GP (r < 1)     S =a /  (1 -r)

Definition of Harmonic Progression (H.P) Series

Harmonic progression is the series when the reciprocal of the terms are in AP. 

For example, 1/a, 1/ (a + d), 1/ (a + 2d)…… are termed as a harmonic progression as a, a + d, a + 2d are in arithmetic progression. 

Harmonic Progression (H.P) series Formula

• The nth term in HP is identified by, T_n =1/ [a + (n -1) d]
• To solve any problem in harmonic progression, a series of AP should be formed first, and then the problem can be solved. 
• For two terms ‘a’ and ‘b’,
  Harmonic Mean = (2 a b) / (a + b)

Relationship Between Arithmetic Mean, Harmonic Mean, and Geometric Mean of Two Numbers

If GM, AM and HM are the Geometric Mean, Arithmetic Mean and Harmonic Mean of two positive numbers respectively, then 

GM² = AM * HM

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