Sequence and series ( A.P & G.P)

An arithmetic progression (AP) is a sequence where the differences between every two consecutive terms are the same. For example, the sequence 2, 6, 10, 14, … is an arithmetic progression (AP) because it follows a pattern where each number is obtained by adding 4 to the previous term.

For example, 1, 5, 9, 13, 17, 21, 25, 29, 33, ... is an arithmetic progression as the differences between every two consecutive terms are the same (as 4). i.e., 5 - 1 = 9 - 5 = 13 - 9 = 17 - 13 = 21 - 17 = 25 - 21 = 29 - 25 = 33 - 29 = ... = 4. We can also notice that every term (except the first term) of this AP is obtained by adding 4 to its previous term. In this arithmetic progression:

a = 1 (the first term)

d = 4 (the "common difference" between terms)

Thus, an arithmetic progression, in general, can be written as: {a, a + d, a + 2d, a + 3d, ... }.

Arithmetic Progression Formula (AP Formulas)

For the first term 'a' of an AP and common difference 'd', given below is a list of arithmetic progression formulas that are commonly used to solve various problems related to AP:

Common difference of an AP:

d = a₂ - a₁ = a₃ - a₂ = a₄ - a₃ = ... = a_n - a_n-1

nth term of an AP: an = a + (n - 1)d

Sum of n terms of an AP: Sn =n/2 (2a + (n - 1) d) = n/2 (a + l), where l is the last term of the arithmetic progression.

Geometric Progression (GP) is a type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, which is called a common ratio. This progression is also known as a geometric sequence of numbers that follow a pattern. . The common ratio multiplied here to each term to get the next term is a non-zero number. An example of a Geometric sequence is 2, 4, 8, 16, 32, 64, …, where the common ratio is 2.

It is represented by:

a, ar, ar2, ar3, ar4, and so on.

Where a is the first term and r is the common ratio.

Note: It is to be noted that when we divide any succeeding term from its preceding term, then we get the value equal to the common ratio.

Suppose we divide the 3rd term by the 2nd term we get:

ar²/ar = r

In the same way:

ar³/ar² = r

ar⁴/ar³ = r

Properties of Geometric Progression (GP)

Some of the important properties of GP are listed below:

Three non-zero terms a, b, c are in GP if and only if b² = ac

In a GP,
Three consecutive terms can be taken as a/r, a, ar
Four consecutive terms can be taken as a/r³, a/r, ar, ar³
Five consecutive terms can be taken as a/r², a/r, a, ar, ar²

Geometric Progression Formulas

The list of formulas related to GP is given below which will help in solving different types of problems.

•The general form of terms of a GP is a, ar, ar², ar³, and so on. Here, a is the first term and r is the common ratio.

•The nth term of a GP is Tn = ar^n-1

•Common ratio = r = T_n/ T_n-1

•The formula to calculate the sum of the first n terms of a GP is given by:
Sn = a[(rⁿ – 1)/(r – 1)] if r ≠ 1and r > 1
Sn = a[(1 – rⁿ)/(1 – r)] if r ≠ 1 and r < 1