**GETTING STARTED**

☛ Progressions are of three types – Arithmetic, Geometric and Harmonic

☛ Adjacent terms in **Arithmetic Progression(AP)** have constant difference between them

**eg. 1,4,7,10,13,16**

The above series with 6 terms is an example of AP.

Let us see the difference between the terms.

4-1 = 3

7-4 = 3

10-7 = 3

13-10 = 3

**The difference is always 3. Hence the terms are in A.P**

**Note:** The difference can also be negative. If the difference is negative, it will be an A.P where the values of terms will be decreasing

**☛ **Adjacent terms in** Geometric Progression **have a constant ratio between them

**eg. 3,6,12,24,48,96**

The above series has 6 terms.

Ratio between terms:

6/3 = 2

12/6 = 2

24/12 =2

48/24 = 2

96/48 = 2

**The ratio between the terms is always 2 in this case. Hence terms are in G.P**

**Note: **The ratio can also be a fraction and less than 1. If the ratio is less than 1, it will be a G.P where the values of terms will be decreasing

**☛ **If reciprocals of terms of a series are in AP, then the terms are said to be in **Harmonic progression (H.P)**

**eg.** **1,1/3,1/5,1/7,1/9,1/11**

The above series has 6 terms. The reciprocals of the terms are: 1,3,5,7,9,11 and they are in A.P

Hence the series is in H.P

## ARITHMETIC PROGRESSION:

**☛FINDING n ^{th}**

**term of A.P**

Let us consider an arithmetic progression where the first term is a and the common difference is d.

The A.P will be a, a+d, a+2d….

nth term of the series will be

**[1st Term]+Interval x Difference **

n terms in sequence will have (n-1) intervals.

Hence,

**nth Term will be**

** [ a + (n-1) . d] **

**Specialty of A.P:**

The **sum of** **first term and last term** is **same as** **sum of second term and second last term** which is **same as sum of third and third last term** and so on, till the central value is reached.

** eg. Let’s consider a series:**

**1,5,9,13,17,21, 25**

**☛**In this series there are **7 terms** which is **odd**, let us check how the above property works.

The sum of end terms

= 1+ 25 = 26

The sum of second and second last terms

= 5 + 21 = 26

The sum of third and third last terms

= 9 + 17 = 26

The sum of fourth and fourth last term (both are same)

= 13 + 13 = 26

If we find the **average of the series, it is 13 and middle term in the series**.

** eg 2. Let us consider another series with even number of terms:**

**1,4,7,10,13,16**

a = 1 and d= 3

**☛**The sum of first and last term = 1+ 16 = 17

The sum of second and second last term = 4 + 13 = 17

The sum of third and third last term = 7 + 10 = 17

If we find **average of the series, it is 8.5**. Though it is **not in the series**, it is the central value between the central values of the A.P

The central values of AP are 7 and 10. Average of the two terms is 8.5 and hence is average of the series.

**☛ **Finding sum of A.P

Sum of A. P = No. of terms of series X Average Value of Series

Average Value of series = (First Term + Last Term)/2

= (a+ [a+(n-1)d]) /2

= [2a + (n-1) d]/2

**Therefore, Sum of A.P **

**= n/2 x [2a+(n-1).d]**

**HACK TO SOLVE A.P Problems:**

## ☛ If the **number of terms in A.P is even, say 4**

Take the terms as a-3d, a-d, a+d and a+3d

The **central term of the four terms is a and the common difference is 2d.**

The reason for that is that the sum of all terms will be 4a.

**☛ **Similarly, **if number of terms in A.P is odd, say 5**

Take the terms as a-2d, a-d, a, a+d and a+2d

The **central term of the four terms is a and the common difference is d.**

The reason for that is that the sum of all terms will be 5a.

## GEOMETRIC PROGRESSION

Let the first term of G.P be a and the common ratio be r.

The terms of G.P, hence, are: a , ar, ar^{ 2 }, ar^{3 }…

**☛ ****nth term of G.P = a . r **^{n-1}

^{n-1}

**☛ ****Sum of terms in G.P:**

**Sum of terms in G.P **

**= a . (r ^{ n} – 1)/(r-1) , for r > 1**

**= a.(1- r ^{ n} )/(1 -r), for -1< r < 1**

The sum can also be expressed as

**(r.[Last Term]-[1st Term])/(r-1)**

**☛ **Geometric mean of G.P is **nth root of the product of all terms in the series**

**HACK TO SOLVE G.P Problems:**

**☛ **If the number of terms in G.P is even, say 4

Take the terms as a/d^{3} , a/d, ad and ad^{3}

The central term of the four terms is a and the common ratio is d^{2}

The reason for that is that the product of all terms will be a^{4} and geometric mean will be a.

**☛ **Similarly, if number of terms in G.P is odd, say 5

Take the terms as a/d^{2}, a/d, a, a.d and a.d^{2}

The central term of the five terms is a and the common ratio is d.

The reason for that is that the product of all terms will be a^{5} and the Geometric mean will be a.

**☛ ****Infinite G.P**

Let us consider a G.P

1, 1/2,1/4,1/8…..

The series is an infinite series because it **does not terminate at all**.

The sum of such a series will not be infinite but will converge at a particular value as the terms will keep on diminishing.

**Sum of infinite G.P is given by a/(1-r), where -1< r < 1**

For common ratio, r > 1, the sum will be infinite.

Eg. 1,2,4,8…

r is 2 in this case and hence the sum is infinite

Harmonic Progression

**☛ **If a, b, c are in harmonic progression, b is said to be the harmonic mean of a and c.

**☛ **In general, if x_{1}, x_{2}, ……x_{n} are in harmonic progression,

**x _{2}, x_{3},….x_{n – 1 }are the (n-2) harmonic means between x_{1} and x_{n}**

**☛ ****Calculating Harmonic Mean of Two Numbers**

HM of two numbers let us say a and b is given as 2a.b/(a+b)

## PROPERTY OF NUMBERS :

For any two positive numbers,

A.M ≥ G.M ≥ H.M