# Study Notes on Sequence & Series

By Gaurav Mohanty|Updated : December 15th, 2021

### Arithmetic Progression

Arithmetic Progression (AP) is defined as a series in which a difference between any two consecutive terms is constant throughout the series. This constant difference is called the common difference. If a is the first term and d is a common difference then an AP can be written as a, (a +d), (a+2d) ...

The nth term of an AP is given by Tn= a + (n-1) d

And the sum up to n terms is given by S= (n/2) [ 2a + (n-1) d] =(n/2) [ a + l] where l is the last or the nth term of the A.P.

Properties of A.P. Series

(a) If a1, a2….., an are in AP with common difference d, then a1 + k, a2 + k, …….., an + k will also be in AP with the same common difference.

(b) If a1, a2,…….., an are in AP with common difference d, then a1 - k, a2 - k, …….., an - k will also be in AP with the same common difference.

(c) If a1, a2,…….., an are in AP with common difference d, then ka1, ka2,…….., kan will also be in AP with the same common difference kd.

(d) If a1, a2,…….., an are in AP with common difference d, then (a1 / k), (a2 / k),……..,

(e) ( an / k),…….., will also be in AP with the same common difference d/k

(f) If a1 , a2,…….., an are in AP with common difference d1 and b1 , b2,…….., bn are in AP with common difference d2 then a1 ± b1, a2 ± b2, a3 ± b3 , ........ an ± bn , is in AP with common difference d1± d2

(g) In a finite AP, the sum of terms equidistant from the beginning and end is always same i.e. a1 + an = a2 + an-1 = …………

(h) Three numbers a, b, c is in AP implies 2b = a + c

(i) If a, A1, A2 ………..., An, b are in AP, then A1, A2 ……….., An are called n Arithmetic Means between a and b.

### Selection of terms in AP

(a) 3 terms: a - d, a, a + d

(b) 4 terms: a-3d, a-d, a + d, a+3d

(c) 5 terms: a-2d, a-d, a, a + d, a+2d

(d) 6 terms: a-5d, a-3d, a-d, a + d, a+3d, a+5d

Sum of first n natural numbers: 1+2+3+…n = [ (n(n+1)) /2]

Sum of squares of 1st n natural numbers: 12 + 22 + 32+……n2 =[ (n(n+1) (2n+1))/6]

Sum of cubes of 1st n natural numbers: 13 + 23 + 33+……n3 = [ (n(n+1)) /2 ]2

### 2. Geometric Progression

GP is defined as a series in which ratio between any two consecutive terms is constant throughout the series. This constant is called the common ratio. If a is the first term and r is the common ratio, then a GP can be written as a, ar, ar2 , ar3 ,………arn

The nth term is given by Tn= ar (n-1)

Sum up to n terms is given by

### Properties of GP

(a) If a1, a2…..., an are in GP with common ratio r then ka1 , ka2,…….., kan will also be in G.P. with the same common ratio provided k is non-zero

(b) If a1, a2…..., an are in GP with common ratio r then (a1 / k), (a2 / k)…..,( an / k) will also be in GP with the same common ratio provided k is non-zero

(c) If a1, a2…..., an are in GP then a1k, a2k…..., ank will also be in GP with common ratio rk

(d) If a1, a2…..., an be in GP with all terms positive and common ration r then log a1, log a2, .…..., log an will be in AP with common difference log r .

(e) The product of two individual GP’s will have common ration as the product of two common ratios

(f) In a finite GP, the product of the terms quidistant from the beginning and the end is always the same i.e. a1an = a2an-1 = ……...

(g) Three numbers a, b, c are in GP if b2 = ac

(h) If a, G1, G2 ………..., Gn , b are in GP, then G1 ,G2 ……….., Gn are called the n geometric means between a and b

### Selection of terms in GP

(a) 3 terms: (a / r), a, ar

(b) 4 terms: (a / r3), (a / r) , ar, ar3

(c) 5 terms: (a / r2), (a / r) ,a, ar, ar2

(d) 6 terms: (a / r5), (a / r3) , a / r, ar, ar3, ar5

### Infinite GP Series

If the number of terms of a GP is very large i.e. n tends to infinite, then such a GP series is known as infinite GP series. An infinite GP will have finite sum if and only if |r|<1

Sum of infinite series = a / (1 - r)

### 3. Arithmetic-Geometric Progression

If each term of a progression is the product of the corresponding terms of an AP and a GP, then it is called a AGP.

i.e. a, (a + d) r, (a + 2d)r2 , (a + 3d) ar3 ,………(a + (n-1)d) rn-1

The general term is Tn= (a + (n-1) d) r (n-1)

To find the sum of n terms we suppose its sum is, Sn . Multiply both sides by the common ratio of GP.

Sn = a + (a + d)r + (a + 2d)r2 + (a + 3d) ar3 + ………+ (a + (n-1)d)rn-1

rSn = ar + (a + d)r2 + (a + 2d)r3 + (a + 3d) ar4 + ………+ (a + (n-1)d)rn

So,

Sn (1 – r) = a + rd + r2d + …… + [ a + (n - 1) d]rn

Sn = (a/(1-r)) + [ rd (1 – rn-1) / (1 – r)2] - [ a + (n - 1) d]rn / (1 – r)

### 4. Harmonic Progression

If a1 , a2,…….., an are in AP such that none of them is 0, then (1/a1 ), (1/a2), …….., (1/an ) are said to be in HP

(a) If a, b, c are in HP, then (1/a), (1/b), (1/c ) are in AP

(b) If a, H1, H2 ………..., Hn, b are in HP, then H1 ,H2 ……….., Hn are called n harmonic means between a and b.

### 5. Inequalities

Arithmetic mean(A) =

Geometric mean(G)=

Harmonic mean(H)=

A≥G≥H

Equality holds when all terms are equal

Weighted arithmetic mean(A*) =

Weighted geometric mean(G*) =

Weighted harmonic mean(H*) =

Equality holds when all the entries are equal

Let x1 , x2 , ...... xn be n positive real numbers and let m be a real number. Then

, if m belongs to all real numbers outside the range [0,1]

, if m lies in between 0 and 1

, if m is equal to 0 or 1

===========================

The most comprehensive exam prep app.

If you are aiming to crack IPM and other BBA Exam, join BYJU'S Exam Prep Online Classroom Program where you get :

• Live Courses by Top Faculty
• Daily Study Plan
• Comprehensive Study Material
• Latest Pattern Test Series
• Complete Doubt Resolution
• Regular Assessments with Report Card

#DreamStriveSucceed