Sequence and Series Notes for NDA exam

1. 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 T_n= a + (n-1) d

And the sum up to n terms is given by S_n = (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….., a_n are in AP with common difference d, then a₁ + k, a₂ + k, …….., a_n + k will also be in AP with the same common difference.

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

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

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

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

(f) If a₁ , a₂,…….., a_n are in AP with common difference d₁ and b₁ , b₂,…….., b_n are in AP with common difference d₂ then a₁ ± b₁, a₂ ± b₂, a₃ ± b₃ , ........ a_n ± b_n , 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 + a_n = a₂ + a_n-1 = …………

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

(i) If a, A₁, A₂ ………..., A_n, b are in AP, then A₁, A₂ ……….., A_n 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 1^st n natural numbers: 1² + 2² + 3²+……n² =[ (n(n+1) (2n+1))/6]

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

2. Geometric Progression

GP is defined as a series in which ration 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 T_n= ar ^(n-1)

Sum up to n terms is given by

Properties of GP

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

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

(c) If a₁, a₂…..., a_n are in GP then a₁^k, a₂^k…..., a_n^k will also be in GP with common ratio r^k

(d) If a₁, a₂…..., a_n be in GP with all terms positive and common ration r then log a₁, log a₂, .…..., log a_n 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. a₁a_n = a₂a_n-1 = ……...

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

(h) If a, G₁, G₂ ………..., G_n , b are in GP, then G₁ ,G₂ ……….., G_n 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) r^n-1

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

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

S_n = a + (a + d)r + (a + 2d)r² + (a + 3d) ar³ + ………+ (a + (n-1)d)r^n-1

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

So,

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

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

4. Harmonic Progression

If a₁ , a₂,…….., a_n are in AP such that none of them is 0, then (1/a₁ ), (1/a₂), …….., (1/a_n ) 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 ………..., H_n, b are in HP, then H₁ ,H₂ ……….., H_n 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 , ...... x_n 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

All the best!
Team BYJU'S Exam Prep