Arithmetic progression and Geometric progression Calculator

Arithmetic Progression

First Term

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Second Term

Common Difference

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Value Of nth term

Number Of Term

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Last Term

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Sum Of term

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Geometric Progression

First Term

Find the First term;

Second Term

Common Ratio

Find the common ratio;

Number Of Term

Find the number of term;

nth Term

Sum Of term

Find the sum of term;

Sum of infinite term

Arithmetic Progression

Arithmetic Progression (AP) is a sequence of numbers in order, in which the difference between any two consecutive numbers is a constant value. It is also called Arithmetic Sequence. For example, the series of natural numbers: 1, 2, 3, 4, 5, 6,… is an Arithmetic Progression, which has a common difference between two successive terms (say 1 and 2) equal to 1 (2 -1). Even in the case of odd numbers and even numbers, we can see the common difference between two successive terms will be equal to 2.

Arithmetic Progression Formulas

There are two major formulas we come across when we learn about Arithmetic Progression, which is related to:

The nth term of AP
Sum of the first n terms

Let us learn here both the formulas with examples.

nth Term of an AP

The formula for finding the n-th term of an AP is:
an = a + (n − 1) × d
Where

a = First term

d = Common difference

n = number of terms

a_n = nth term

Example: Find the nth term of AP: 1, 2, 3, 4, 5…., an, if the number of terms are 15.

Solution: Given, AP: 1, 2, 3, 4, 5…., an
n=15
By the formula we know, an = a+(n-1)d
First-term, a =1
Common difference, d=2-1 =1
Therefore, a_n = a ₁₅ = 1+(15-1)1 = 1+14 = 15

Note: The behaviour of the sequence depends on the value of a common difference.
If the value of “d” is positive, then the member terms will grow towards positive infinity
If the value of “d” is negative, then the member terms grow towards negative infinity

Sum of N Terms of AP

For an AP, the sum of the first n terms can be calculated if the first term, common difference and the total terms are known. The formula for the arithmetic progression sum is explained below:

Consider an AP consisting “n” terms.
Sn = n/2[2a + (n − 1) × d]
This is the AP sum formula to find the sum of n terms in series.

Sum of AP when the Last Term is Given

Formula to find the sum of AP when first and last terms are given as follows:
S = n/2 (first term + last term)

General Form of AP	a, a + d, a + 2d, a + 3d, . . .
The nth term of AP	a_n = a + (n – 1) × d
Sum of n terms in AP	S = n/2[2a + (n − 1) × d]
Sum of all terms in a finite AP with the last term as ‘l’	n/2(a + l)

Geometric Sequence /Progression

A geometric progression or a geometric sequence is the sequence, in which each term is varied by another by a common ratio. The next term of the sequence is produced when we multiply a constant (which is non-zero) to the preceding term. 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/a³ = 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,

In a finite GP, the product of the terms equidistant from the beginning and the end is the same

ⁿ

If each term of a GP is multiplied or divided by a non-zero constant, then the resulting sequence is also a GP with the same common ratio
The product and quotient of two GP’s is again a GP
If each term of a GP is raised to the power by the same non-zero quantity, the resultant sequence is also a GP
If a1, a2, a3,… is a GP of positive terms then log a1, log a2, log a3,… is an AP (arithmetic progression) and vice versa

General Form of Geometric Progression

The general form of Geometric Progression is:

a, ar, ar², ar³, ar⁴,…, arn-1

Where,

a = First term

r = common ratio

ar^n-1 = nth term

General Term or Nth Term of Geometric Progression

Let a be the first term and r be the common ratio for a Geometric Sequence.

Then, the second term, a₂ = a × r = ar

Third term, a₃ = a₂× r = ar × r = ar²

Similarly, nth term, a _n = ar^n-1

Therefore, the formula to find the nth term of GP is:

an = tn = arn-1

Note: The nth term is the last term of finite GP.

Common Ratio of GP

Consider the sequence a, ar, ar², ar³,……

First term = a

Second term = ar

Third term = arⁿ

Similarly, nth term, t_n= ar^n-1

Thus, the common ratio of geometric progression formula is given as:

Common ratio = (Any term) / (Preceding term)

= tⁿ / t^n-1

= (ar ^n-1) /(ar ^n-2)

= r
Thus, the general term of a GP is given by arn-1 and the general form of a GP is a, ar, ar²,…..

For Example: r = t2 / t1 = ar / a = r

Sum of N term of GP

Suppose a, ar, ar², ar³,……ar^n-1 is the given Geometric Progression.

Then the sum of n terms of GP is given by:

Sn = a + ar + ar² + ar³ +…+

The formula to find the sum of n terms of GP is:

Sn = a[(rⁿ – 1)/(r – 1)] if r ≠ 1 and r > 1

Where

a is the first term

r is the common ratio

n is the number of terms

Also, if the common ratio is equal to 1, then the sum of the GP is given by:

Sn = na if r = 1

Types of Geometric Progression

Geometric progression can be divided into two types based on the number of terms it has. They are:

Finite geometric progression (Finite GP)

Infinite geometric progression (Infinite GP)

These two GPs are explained below with their representations and the formulas to find the sum.

Finite Geometric Progression
The terms of a finite G.P. can be written as a, ar, ar², ar³,……ar^n-1is called finite geometric series.
The sum of finite Geometric series is given by:
S_n = a[(r ⁿ – 1)/(r – 1)] if r ≠ 1 and r > 1

Infinite Geometric Progression

Terms of an infinite G.P. can be written as a, ar, ar², ar³, ……ar^n-1,…… is called infinite geometric series.
The sum of infinite geometric series is given by:

S _n = a / 1 - r

This is called the geometric progression formula of sum to infinity.

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 = arn-1
Common ratio = r = Tn/ Tn-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
The nth term from the end of the GP with the last term l and common ratio r = l/ [r(n – 1)].
The sum of infinite, i.e. the sum of a GP with infinite terms is S∞= a/(1 – r) such that 0 < r < 1.
If three quantities are in GP, then the middle one is called the geometric mean of the other two terms.
If a, b and c are three quantities in GP, then and b is the geometric mean of a and c. This can be written as b2 = ac or b =√ac
Suppose a and r be the first term and common ratio respectively of a finite GP with n terms. Thus, the kth term from the end of the GP will be = arn-k.