👉    A sequence is a succession of quantities, each of which is formed according to a definite law.
e.g.: (a) 3, 6, 9, 12, … (b) 4, 16, 64, 256, …

👉    If the sequence has a last term, it is finite; otherwise, it is infinite.

👉    The algebraic sum of a succession of quantities, each formed according to a definite rule or law, is called a series.
e.g.: (a) 3 + 6 + 9 + 12 + … (b) 4 + 4² + 4³ + 4⁴ + …

👉    A sequence of numbers is said to be a progression when the difference or ratio of any term and its preceding term is constant throughout the whole sequence.

👉    A sequence of quantities is said to be in arithmetic progression (A.P.) if the difference of any term and its preceding term is constant throughout the whole sequence.

👉    The constant difference obtained by subtracting any term and its preceding term is called the common difference.

👉    If a sequence is in arithmetic progression, its corresponding series is called an arithmetic series (A.S.).

👉    n^th term of an A.P. (t_n​) = a+(n−1)d, where aa is the first term (t₁​).

👉    Sum of the first n terms of an A.P.  $S_n=\frac{n}{2}[2a+(n-1)d]$

👉    A sequence of quantities is said to be in harmonic progression (H.P.) if their reciprocals form an arithmetic progression.

👉    If a sequence is in harmonic progression, its corresponding series is called a harmonic series (H.S.).

👉    A sequence of quantities is said to be in geometric progression (G.P.) if the ratio of each term to its preceding one is the same throughout the whole sequence.

👉    The constant number obtained by dividing any term by the preceding term is called the common ratio (r).

👉    If a sequence is in geometric progression, its corresponding series is called a geometric series (G.S.).

👉    n^th term of a G.P. (t_n​) = ar^n−1.

👉    Sum of the first n terms of a G.P. 

$S_n=\frac{a(r^n-1)}{r-1},\ r>1$

$S_n=\frac{a(1-r^n)}{1-r},\ r$ 

👉    Sum of an infinite geometric series $S_\infty=\frac{a}{1-r},\ |r|$​

👉    If three numbers are in A.P., then the middle term is called the arithmetic mean of the first and third terms.

👉    Arithmetic mean (A.M.) between a and b, when n arithmetic means are inserted:

       $M_n = a + \frac{n(b-a)}{n+1}$