CBSE Class 10 Arithmetic Progression Important Formulas and concepts for exams

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CLASS X : CHAPTER - 5
ARITHMETIC PROGRESSION (AP)
IMPORTANT FORMULAS & CONCEPTS

SEQUENCE

An arrangement of numbers in a definite order according to some rule is called a sequence. In other words, a pattern of numbers in which succeeding terms are obtained from the preceding term by adding/subtracting a fixed number or by multiplying with/dividing by a fixed number, is called sequence or list of numbers.
e.g. 1, 2, 3, 4, 5

A sequence is said to be finite or infinite accordingly it has finite or infinite number of terms. The various numbers occurring in a sequence are called its terms.

ARITHMETIC PROGRESSION (AP)

An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.
This fixed number is called the common difference of the AP. It can be positive, negative or zero.
Let us denote the first term of an AP by \(a_1\), second term by \(a_2\), \(\ldots\), nth term by \(a_n\) and the common difference by \(d\). Then the AP becomes \(a_1, a_2, a_3, \ldots, a_n\).
So, \(a_2 - a_1 = a_3 - a_2 = \ldots = a_n - a_{n-1} = d\).
The general form of an arithmetic progression is given by
\[a,\ a + d,\ a + 2d,\ a + 3d,\ \ldots\] where \(a\) is the first term and \(d\) the common difference.

nth Term of an AP

Let \(a_1, a_2, a_3, \ldots\) be an AP whose first term \(a_1\) is \(a\) and the common difference is \(d\).
Then,
the second term \(a_2 = a + d = a + (2 - 1)d\)
the third term \(a_3 = a_2 + d = (a + d) + d = a + 2d = a + (3 - 1)d\)
the fourth term \(a_4 = a_3 + d = (a + 2d) + d = a + 3d = a + (4 - 1)d\)
\(\ldots\ldots\ldots\)
\(\ldots\ldots\ldots\)
Looking at the pattern, we can say that the nth term \(a_n = a + (n - 1)d\).
So, the nth term \(a_n\) of the AP with first term \(a\) and common difference \(d\) is given by

\[a_n = a + (n - 1)\,d\]

\(a_n\) is also called the general term of the AP. If there are \(m\) terms in the AP, then \(a_m\) represents the last term which is sometimes also denoted by \(l\).

nth Term from the end of an AP

Let the last term of an AP be \(l\) and the common difference of an AP be \(d\). Then the nth term from the end of an AP is given by

\[l_n = l - (n - 1)\,d\]

Sum of First n Terms of an AP

The sum of the first \(n\) terms of an AP is given by

\[S_n = \frac{n}{2}\left[2a + (n - 1)d\right]\]

where \(a\) = first term, \(d\) = common difference and \(n\) = number of terms.

Also, it can be written as

\[S_n = \frac{n}{2}\left[a + a_n\right]\]

where \(a_n\) = nth term

or

\[S_n = \frac{n}{2}\left[a + l\right]\]

where \(l\) = last term.
This form of the result is useful when the first and the last terms of an AP are given and the common difference is not given.

Sum of first \(n\) positive integers is given by \(S_n = \dfrac{n(n + 1)}{2}\)

Problems based on finding \(a_n\) if \(S_n\) is given.
Find the nth term of the AP, follow the steps:

1. Consider the given sum of first \(n\) terms as \(S_n\).
2. Find the value of \(S_1\) and \(S_2\) by substituting the value of \(n\) as 1 and 2.
3. The value of \(S_1\) is \(a_1\) i.e. \(a\) = first term and \(S_2 - S_1 = a_2\).
4. Find the value of \(a_2 - a_1 = d\), common difference.
5. By using the value of \(a\) and \(d\), write AP.

Problems based on finding \(S_n\) if \(a_n\) is given.
Find the sum of \(n\) term of an AP, follow the steps:

1. Consider the nth term of an AP as \(a_n\).
2. Find the value of \(a_1\) and \(a_2\) by substituting the value of \(n\) as 1 and 2.
3. The value of \(a_1\) is \(a\) = first term.
4. Find the value of \(a_2 - a_1 = d\), common difference.
5. By using the value of \(a\) and \(d\), write AP.
6. By using \(S_n\) formula, simplify the expression after substituting the value of \(a\) and \(d\).

Arithmetic Mean
If \(a\), \(b\) and \(c\) are in AP, then \(b\) is known as arithmetic mean between \(a\) and \(c\)

\[b = \frac{a + c}{2}\]

i.e. AM between \(a\) and \(c\) is \(\dfrac{a + c}{2}\).

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