Arithmetic Progression (AP) **Nth Term, AP Formula** is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is known as the common difference.

The nth term of an arithmetic progression can be calculated using the formula:

**a_n = a_1 + (n-1)d**

where a_n is the nth term, a_1 is the first term, n is the term number, and d is a common difference.

The sum of the first n terms of an arithmetic progression can be calculated using the formula:

**S_n = n/2[2a_1 + (n-1)d]**

where S_n is the sum of the first n terms, a_1 is the first term, n is the number of terms, and d is a common difference.

For example, let’s consider an arithmetic progression with a_1 = 2 and d = 3.

The first five terms of this sequence would be:

2, 5, 8, 11, 14

To find the nth term, we can use the formula:

**a_n = a_1 + (n-1)d**

Let’s say we want to find the 10th term:

a_10 = 2 + (10-1)3 a_10 = 2 + 27 a_10 = 29

So the 10th term of this arithmetic progression is 29.

To find the sum of the first n terms, we can use the formula:

S_n = n/2[2a_1 + (n-1)d]

Let’s say we want to find the sum of the first 7 terms:

S_7 = 7/2[2(2) + (7-1)3] S_7 = 7/2[4 + 18] S_7 = 7/2[22] S_7 = 77

So the sum of the first 7 terms of this arithmetic progression is 77.

**What is Arithmetic Progression**?

Arithmetic Progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a fixed constant value to the preceding term. The constant value added to each term is called the common difference (d).

For example, the sequence 2, 5, 8, 11, 14 is an arithmetic progression, where the first term is 2 and the common difference is 3. Each term is obtained by adding 3 to the preceding term.

In general, the nth term (a_n) of an arithmetic progression with the first term (a_1) and common difference (d) is given by the formula:

a_n = a_1 + (n – 1) d

Arithmetic Progressions are used in various fields such as mathematics, physics, engineering, and finance to model linear relationships and trends. They also play a significant role in number theory and algebra.

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**Notation in Arithmetic Progression**

n is the number of terms [for example 1, 4, 7 has 3 terms where (1) is the first term (4) is the second, and (7) is the third term of the sequence]

a_{1} is the first term

T_{n} is the n-th term

d is the common difference

S_{n} is the sum of n number of terms in arithmetic progression

The n-th term of AP is written as, T_{n} = a + (n-1)d

The sum of n number of terms of AP is written as, S_{n} = n/2 [2a + (n-1)]d

**Arithmetic Progression Question and Answer**

Arithmetic Progression, commonly abbreviated as AP, is a type of sequence in which the difference between consecutive terms is constant. In other words, if you add or subtract a fixed number from one term to get to the next term, you have an arithmetic progression.

In this Q&A session, we will be solving various types of arithmetic progression problems, including finding the nth term, the sum of the first n terms, finding the common difference, and more. By the end of this session, you should have a good understanding of the key concepts of arithmetic progression and be able to solve different types of problems related to it.

**1. ****The second and the sixth terms of an A.P. are 10 and 34. Find the (a) common difference (b) first term (c) nth term (d) sum of first 100 terms**

Let the first term of the arithmetic progression be a, and let d be a common difference. Then we have:

second term = a + d = 10 sixth term = a + 5d = 34

Subtracting the first equation from the second, we get:

4d = 24

So, d = 6.

Substituting this value of d in the first equation, we get:

a + 6 = 10

So, a = 4.

(a) The common difference is 6. (b) The first term is 4. (c) The nth term of an arithmetic progression with first term a and common difference d is given by:

an = a + (n-1)d

So, in this case, we have:

an = 4 + (n-1)6 = 6n – 2

(d) The sum of the first n terms of an arithmetic progression with first term a and common difference d is given by:

Sn = (n/2)(2a + (n-1)d)

So, in this case, we have:

S100 = (100/2)(2(4) + (100-1)6) = 50(2(4) + 99(6)) = 50(596) = 29800

Therefore, the sum of the first 100 terms is 29800.

**2. The 3rd term of an arithmetic progression is 8 and the 9th term is 32. Find the common difference and the first term.**

Let the first term of the arithmetic progression be ‘a’ and the common difference be ‘d’.

Then, the third term can be expressed as a + 2d = 8 ….(1)

Similarly, the ninth term can be expressed as a + 8d = 32 ….(2)

We can solve these two equations simultaneously to find the values of ‘a’ and ‘d’.

Subtracting equation (1) from equation (2), we get:

6d = 24

d = 4

Substituting this value of ‘d’ in equation (1), we get:

a + 2(4) = 8

a + 8 = 8

a = 0

Therefore, the first term of the arithmetic progression is 0 and the common difference is 4.

**3. Find the sum of the first 20 terms of an arithmetic progression whose first term is 3 and common difference is 4.**

The formula for the sum of the first n terms of an arithmetic progression is:

S_n = n/2 [2a + (n-1)d]

Where a is the first term, d is a common difference, and n is the number of terms.

Substituting the given values, we get:

S_20 = 20/2 [2(3) + (20-1)(4)]

S_20 = 10 [6 + 76]

S_20 = 820

Therefore, the sum of the first 20 terms of the given arithmetic progression is 820.

**4. Find the nth term of an arithmetic progression whose first term is 2 and common difference is 5.**

The nth term of an arithmetic progression can be found using the formula:

a_n = a_1 + (n-1)d

Where a_1 is the first term, d is a common difference, and n is the term number.

Substituting the given values, we get:

a_n = 2 + (n-1)5

a_n = 2 + 5n – 5

a_n = 5n – 3

Therefore, the nth term of the given arithmetic progression is 5n – 3.

**5. If the sum of the first n terms of an arithmetic progression is given by Sn = 3n^2 – 5n, find the first term and the common difference of the progression.**

To solve the problem, we will use the formula for the sum of an arithmetic progression:

Sn = n/2[2a + (n-1)d],

where a is the first term, d is a common difference, and n is the number of terms.

From the given formula, Sn = 3n^2 – 5n, we can substitute Sn with the given value and simplify:

3n^2 – 5n = n/2[2a + (n-1)d]

6n^2 – 10n = n(2a + (n-1)d)

6n – 10 = 2a + (n-1)d

2a = 6n – 10 – (n-1)d

2a = 5n – d – 10

Now, we will use the formula for the first term of an arithmetic progression:

a = Sn/n – (n-1)d/2

Substituting Sn with the given value and simplifying, we get:

a = (3n^2 – 5n)/n – (n-1)d/2

a = 3n – 5 – (n-1)d/2

Substituting this value of an into the previous equation, we get:

2(3n – 5 – (n-1)d/2) = 5n – d – 10

6n – 10 – (n-1)d = 5n – d – 10

n = d

Hence, the common difference is n.

Now, substituting this value of d into the equation 2a = 5n – d – 10, we get:

2a = 5n – n – 10

2a = 4n – 10

a = 2n – 5

Therefore, the first term of the arithmetic progression is 2n – 5 and the common difference is n.

**CONCLUSION**

An arithmetic progression (AP) is a mathematical sequence where the difference between any two consecutive terms is always a constant value. This constant is also called the common difference, denoted by d, and it remains the same throughout the entire sequence.

The general formula to find the nth term of an arithmetic progression is given:

an = a1 + (n-1)d

where a1 is the first term, d is a common difference, and an is the nth term.

Arithmetic progressions are widely used in various fields, including mathematics, physics, and computer science. They play an essential role in various applications such as calculating financial investments, finding the average rates of change, and predicting future values in data analysis.