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# Arithmetic Progressions Formulas

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## progressions Important Formulas

Progressions are numbers arranged in a particular order such that they form a predictable order. By predictable order, we mean that given some numbers, we can find next numbers in the series.

The progressions can be widely classified into three different types i.e.:

$$\mathbb{1.}$$ Arithmetic Progression
$$\mathbb{2.}$$ Geometric Progression and
$$\mathbb{3.}$$ Harmonic Progression

## 1. Arithmetic Progression Formulas:

An arithmetic progression is a sequence of numbers in which each term is derived from the preceding term by adding or subtracting a fixed number called the common difference "d".The general form of an Arithmetic Progression is a, a + d, a + 2d, a + 3d and so on.

Example, 2,4,6,8,10 is an AP because difference between any two consecutive terms in the series (common difference) is same (4 – 2 = 6 – 4 = 8 – 6 = 10 – 8 = 2).

$$\mathbb{i.}$$ nth term of an AP series is Tn = a + (n - 1) d, where Tn = nth term and a = first term. Here d = common difference = Tn - Tn-1.

$$\mathbb{ii.}$$ Sum of first n terms of an AP: S = (n/2)[2a + (n- 1)d]

$$\mathbb{iii.}$$ The sum of n terms is also equal to the formula Sn = (n/2)(a + l). where l is the last term.

$$\mathbb{iv}$$ Tn = Sn - Sn - 1 , where Tn = nth term

$$\mathbb{v.}$$ When three quantities are in AP, the middle one is called as the arithmetic mean of the other two. If a, b and c are three terms in AP then b = (a+c)/2

## 2. Geometric Progression Formulas:

A sequence of numbers is called a geometric progression in which each term is derived by multiplying or dividing the preceding term by a fixed number called the common ratio.

Example, the sequence 4, -2, 1, - 1/2,.... is a Geometric Progression (GP) for which this -1/2 fixed number is called the common ratio.
The general form of a geometric progression (GP) is a, ar, ar2, ar3 and so on.

$$\mathbb{i.}$$ The nth term of a GP series is Tn = arn-1
where a = first term and r = common ratio = $$\frac{Tn}{(Tn-1)}$$.

$$\mathbb{ii.}$$ The formula applied to calculate sum of first n terms of a GP:
Sn = $$\frac{a(r^{n} - 1)}{(r - 1)}$$

$$\mathbb{iii.}$$ When three quantities are in GP, the middle one is called as the geometric mean of the other two. If a, b and c are three quantities in GP and b is the geometric mean of a and c i.e. b = $$\sqrt{ac}$$

$$\mathbb{iv.}$$ The sum of infinite terms of a GP series S = $$\frac{a}{(1-r)}$$ where 0
$$\mathbb{v.}$$ If a is the first term, r is the common ratio of a finite G.P. consisting of m terms, then the nth term from the end will be = arm-n.

$$\mathbb{vi.}$$ The nth term from the end of the G.P. with the last term l and common ratio r is $$\frac{l}{r^{(n-1)}}$$ .

## 3. Harmonic Progression Formulas:

A sequence of numbers is called a harmonic progression when their reciprocals are in Arithmetic Progression.
In simple terms, a, a + d, a + 2d are in AP then their reciprocals $$\frac{1}{a}$$, $$\frac{1}{a+d}$$, $$\frac{1}{a+2d}$$ will be in HP.

$$\mathbb{i.}$$ The nth term of a HP series is Tn = $$\frac{1}{[a + (n -1)d]}$$.
Note: nth term of H.P. = $$\frac{1}{(nth \ term \ of \ corresponding \ A.P.)}$$. You can convert the HP series in AP and find the nth term, and reciprocal it.

Note: Best approach to solve a problem based on Harmonic Progression, convert the HP serice in corresponding AP series and then solve the problem.

$$\mathbb{ii.}$$ If three terms a, b, c are in HP, then b = $$\frac{2ac}{(a+c)}$$.