# 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 T_{n} = nth term and a = first term. Here d = common difference = T_{n} - T_{n-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 S_{n} = (n/2)(a + l).
where l is the last term.

\( \mathbb{iv}\) T_{n} = S_{n} - S_{n - 1} , where Tn = n^{th} 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, ar^{2}, ar^{3} and so on.

\( \mathbb{i.}\) The n^{th} term of a GP series is T_{n} = ar^{n-1}

where a = first term and r = common ratio = \(\frac{T_{n}}{(T_{n-1})}\).

\( \mathbb{ii.}\) The formula applied to calculate sum of first n terms of a GP:

S_{n} = \(\frac{a(r^{n} - 1)}{(r - 1)}\)

\( \mathbb{iii.}\) 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 = ar^{m-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 n^{th} term of a HP series is T_{n} = \(\frac{1}{[a + (n -1)d]}\).

Note: n^{th} term of H.P. = \(\frac{1}{(nth \ term \ of \ corresponding \ A.P.)}\). You can convert the HP series in AP and find the n^{th} 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 = \(\frac{2ac}{(a+c)}\).