Define an Arithmetic Progression (AP) and explain its general form. Derive the formula for the nth term of an AP and also explain how to find the sum of the first 'n' terms of an AP with proper mathematical justifications.

Q: Define an Arithmetic Progression (AP) and explain its general form. Derive the formula for the nth term of an AP and also explain how to find the sum of the first 'n' terms of an AP with proper mathematical justifications.

Definition of Arithmetic Progression (AP)\nAn Arithmetic Progression is a sequence of numbers in which each term after the first is obtained by adding a fixed number, called the common difference, to the preceding term. This fixed number can be positive, negative, or zero. If the first term is denoted by '$a$' and the common difference by '$d$', the terms of the AP can be written in a specific sequence. General Form of an AP\nThe general form of an Arithmetic Progression is given by: $$a, (a + d), (a + 2d), (a + 3d), \dots$$\nHere, the first term is $a1 = a$, the second term is $a2 = a + d$, the third term is $a3 = a + 2d$, and so on. The difference between any term and its preceding term is always constant and equal to $d$, i.e., $a{k+1} - ak = d$. Derivation of the $n$-th Term ($an$)\nTo find a general expression for any term in an AP, let us analyze the pattern of the terms: First term ($a1$) = $a = a + (1 - 1)d$ Second term ($a2$) = $a + d = a + (2 - 1)d$ Third term ($a3$) = $a + 2d = a + (3 - 1)d$ Fourth term ($a4$) = $a + 3d = a + (4 - 1)d$ \nFollowing this established pattern, for the $n$-th term ($an$), the multiplier of $d$ is one less than the term number $n$. Therefore, the formula for the $n$-th term is: $$an = a + (n - 1)d$$\nwhere $an$ represents the $n$-th term, $a$ is the first term, $n$ is the number of terms, and $d$ is the common difference. Sum of the First '$n$' Terms of an AP ($Sn$)\nLet $Sn$ denote the sum of the first $n$ terms of an AP. The sum can be written as: $$Sn = a + (a + d) + (a + 2d) + \dots + a + (n - 1)d$$ \nRewriting the series in reverse order: $$Sn = a + (n - 1)d + a + (n - 2)d + \dots + a$$\nAdding both equations term by term: $$2Sn = 2a + (n - 1)d + 2a + (n - 1)d + \dots + 2a + (n - 1)d \text{ (n times)}$$ $$2Sn = n \times 2a + (n - 1)d$$ $$Sn = \frac{n}{2} 2a + (n - 1)d$$\nAlternatively, this can be expressed as: $$Sn = \frac{n}{2} a + (a + (n - 1)d) = \frac{n}{2} (a + l)$$\nwhere $l$ is the last term of the AP.

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Definition of Arithmetic Progression (AP)
An Arithmetic Progression is a sequence of numbers in which each term after the first is obtained by adding a fixed number, called the common difference, to the preceding term. This fixed number can be positive, negative, or zero. If the first term is denoted by '$a$' and the common difference by '$d$', the terms of the AP can be written in a specific sequence. General Form of an AP
The general form of an Arithmetic Progression is given by: $$a, (a + d), (a + 2d), (a + 3d), \dots$$
Here, the first term is $a1 = a$, the second term is $a2 = a + d$, the third term is $a3 = a + 2d$, and so on. The difference between any term and its preceding term is always constant and equal to $d$, i.e., $a{k+1} - ak = d$. Derivation of the $n$-th Term ($an$)
To find a general expression for any term in an AP, let us analyze the pattern of the terms: First term ($a1$) = $a = a + (1 - 1)d$ Second term ($a2$) = $a + d = a + (2 - 1)d$ Third term ($a3$) = $a + 2d = a + (3 - 1)d$ Fourth term ($a4$) = $a + 3d = a + (4 - 1)d$ 
Following this established pattern, for the $n$-th term ($an$), the multiplier of $d$ is one less than the term number $n$. Therefore, the formula for the $n$-th term is: $$an = a + (n - 1)d$$
where $an$ represents the $n$-th term, $a$ is the first term, $n$ is the number of terms, and $d$ is the common difference. Sum of the First '$n$' Terms of an AP ($Sn$)
Let $Sn$ denote the sum of the first $n$ terms of an AP. The sum can be written as: $$Sn = a + (a + d) + (a + 2d) + \dots + a + (n - 1)d$$ 
Rewriting the series in reverse order: $$Sn = a + (n - 1)d + a + (n - 2)d + \dots + a$$
Adding both equations term by term: $$2Sn = 2a + (n - 1)d + 2a + (n - 1)d + \dots + 2a + (n - 1)d 	ext{ (n times)}$$ $$2Sn = n 	imes 2a + (n - 1)d$$ $$Sn = \frac{n}{2} 2a + (n - 1)d$$
Alternatively, this can be expressed as: $$Sn = \frac{n}{2} a + (a + (n - 1)d) = \frac{n}{2} (a + l)$$
where $l$ is the last term of the AP.

MP Board · Class 10 · Mathematics · Arithmetic ProgressionsDefine an Arithmetic Progression (AP) and explain its general form. Derive the formula for the nth term of an AP and also explain how to find the sum of the first 'n' terms of an AP with proper mathematical justifications.

General Form of an AP\nThe general form of an Arithmetic Progression is given by:

Derivation of the $n$-th Term ($a_n$)\nTo find a general expression for any term in an AP, let us analyze the pattern of the terms:

Sum of the First '$n$' Terms of an AP ($S_n$)\nLet $S_n$ denote the sum of the first $n$ terms of an AP. The sum can be written as: