Sequence and Series

Sequence

List of terms in a specific order is called a sequence.

E.g. First 10 prime numbers in ascending order.

Series

Sum of terms in a sequence is called a series.

E.g. Sum of first 10 prime numbers.

Progression

Sequence with a specific formula to derive its next term is called a progression.

E.g. First 10 natural numbers in ascending order.

General Sequences and Series

1.	n ∑ r = 1 k	=	k + k + ... n times	=	nk
2.	n ∑ r = 1 r	=	1 + 2 + 3 + ... + n	=	n (n + 1) 2
3.	n ∑ r = 1 r²	=	1² + 2² + ... + n²	=	n (n + 1) (2n + 1) 6
4.	n ∑ r = 1 r³	=	1³ + 2³ + ... + n³	=	n² (n + 1)² 4

Arithmetic Progression (AP)

A progression in which each term is derived by adding a constant to its preceding term is called Arithmetic Progression.

a, a + d, a + 2d, a + 3d, ...

General Term ( t_n ):

t_n = a + (n − 1)d

t_n	: n^th term
a	: First term
d	: Common difference

Sum of n Terms in AP ( S_n ):

S_n	=	n 2 [2a + (n − 1)d]
	=	n 2 (a + t_n)

n	: Number of terms
a	: First term
t_n	: n^th term or last term

Geometric Progression (GP)

A progression in which each term is derived by multiplying a constant with its preceding term is called Geometric Progression.

a, ar, ar², ar³, ...

General Term ( t_n ):

t_n = ar^{n - 1}

t_n	: n^th term
a	: First term
r	: Common ratio

Sum of n Terms in GP ( S_n ):

S_n	=	a (1 − rⁿ) 1 − r , if r ≠ 1
	=	na , if r = 1

Harmonic Progression (HP)

A progression in which each term is derived by taking the reciprocal of its corresponding term in arithmetic progression is called Harmonic Progression.

1 a , 1 a + d , 1 a + 2d , ...

General Term ( t_n ):

t_n = 1 a + (n − 1)d

t_n	: n^th term
a	: First term in AP
d	: Common difference in AP

Sum of n Terms in HP ( S_n ):

S_n = 1 d ln ( 2a + (2n − 1)d 2a − d )