Arithmetic Progressions — Detailed Notes

Chapter 5 | NCERT Class 10 Mathematics

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Arithmetic Progressions — Decision Map

Arithmetic Progressions — Formula Strip

AP Terms on a Number Line Idea

5.1 Introduction

Many natural and daily-life lists follow patterns. This chapter studies the pattern where each term is obtained by adding a fixed number to the previous term — an arithmetic progression (AP). Other patterns (geometric growth, squares, Fibonacci) appear in the intro only as contrast.

5.2 Arithmetic Progressions

Definition

An AP is a list of numbers in which each term (except the first) is obtained by adding a fixed number — the common difference $d$ — to the preceding term. Here $d$ may be positive, negative, or zero.

If $a_{1}, a_{2}, \dots, a_{n}$ is an AP, then

$a_{2} - a_{1} = a_{3} - a_{2} = \dots = a_{n} - a_{n - 1} = d$

In general, $d = a_{k + 1} - a_{k}$ .

General form

$a, a + d, a + 2 d, a + 3 d, \dots$

Finite AP: has a last term $l$ .
Infinite AP: continues without a last term.

To determine an AP you need both first term $a$ and common difference $d$ .

Recognising an AP

Check that consecutive differences are equal. One careful difference is enough once you know it is an AP; to test a list, compare several consecutive differences.

Worked ideas (Example 2 style):

$4, 10, 16, 22, \dots$ differences $6$ $\Rightarrow$ AP; next $28, 34$ .
$- 2, 2, - 2, 2, \dots$ differences alternate $4, - 4$ $\Rightarrow$ not AP.
$1, 1, 1, 2, 2, 2, \dots$ early differences $0$ then $1$ $\Rightarrow$ not AP.

Better Method / Strategy

Always compute $d$ as (next) − (previous), even if the sequence is decreasing.

Situations with compound interest or “remove $1/4$ of remaining air” are typically not APs.

5.3 $n$ th Term of an AP

$a_{n} = a + (n - 1) d$

$a_{n}$ is the general term. If the AP has $m$ terms, $a_{m} = l$ (last term).

Worked ideas

Find a term (Example 3): AP $2, 7, 12, \dots$ $\Rightarrow$ $a = 2$ , $d = 5$ , $a_{10} = 2 + 9 \cdot 5 = 47$ .

Which term / is $0$ a term? (Example 4): $21, 18, 15, \dots$ , find $n$ for $a_{n} = - 81$ : $- 81 = 21 + (n - 1) (- 3) \Rightarrow n = 35$ Also $a_{8} = 0$ .

Two given terms (Example 5): $a_{3} = 5$ , $a_{7} = 9$ $\Rightarrow$ $a + 2 d = 5$ , $a + 6 d = 9$ $\Rightarrow$ $a = 3$ , $d = 1$ .

Is a number a term? (Example 6): Solve for $n$ ; if $n$ is not a positive integer, it is not a term ( $301$ in $5, 11, 17, \dots$ gives $n = 151/3$ ).

Counting pattern (Example 7): Two-digit multiples of $3$ : $12, 15, \dots, 99$ $\Rightarrow$ $n = 30$ .

From the last (Example 8): Prefer reversing: last becomes first, $d$ changes sign; or relate “ $k$ th from last” to the appropriate forward index.

Better Method / Strategy — Finding $n$ vs finding $a_{n}$

Know… Solve…

$a, d, n$ $a_{n}$ directly

$a, d, a_{n}$ $n = \frac{a _{n} - a}{d} + 1$ (check integer)

Two terms $a_{p}, a_{q}$ Linear system in $a, d$

5.4 Sum of First $n$ Terms of an AP

Gauss’s pairing idea yields:

$S_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} (a + a_{n}) = \frac{n}{2} (a + l)$

Use $a + l$ when the last term is known and $d$ is not needed.

Important relation:

$a_{n} = S_{n} - S_{n - 1} (n \geq 2), a_{1} = S_{1}$

Worked ideas

Direct sum (Example 11): $8, 3, - 2, \dots$ , first $22$ terms: $a = 8$ , $d = - 5$ , $S_{22} = \frac{22}{2} [16 + 21 (- 5)] = - 979$

From $S_{n}$ to $a_{n}$ (Example 12): $S_{14} = 1050$ , $a = 10$ $\Rightarrow$ find $d$ , then $a_{20}$ .

How many terms for a given sum (Example 13): May get two admissible $n$ (positive terms cancel later when $d < 0$ ).

Sum of first $n$ positives: $S_{n} = \frac{n ( n + 1 )}{2}$ .

Better Method / Strategy — Which $S_{n}$ form?

Given Prefer

$a, d, n$ $\frac{n}{2} [2 a + (n - 1) d]$

$a, l, n$ $\frac{n}{2} (a + l)$

$S_{n}$ and $S_{n - 1}$ $a_{n} = S_{n} - S_{n - 1}$

Extra standard patterns

Multiples in a range: First multiple $\geq$ lower bound, last multiple $\leq$ upper bound, difference $=$ the multiple base; then find $n$ from $a_{n} = l$ .

Same $d$ , difference of corresponding terms: If two APs share $d$ , then $A_{n} - B_{n} = A_{1} - B_{1}$ (constant for all $n$ ). So difference of 100th terms $=$ difference of 1000th terms.

Term from the end: For an AP with last term $l$ and common difference $d$ , the $k$ th term from the end is $l + (k - 1) (- d) = l - (k - 1) d$ (equivalently reverse the AP).

When $S_{n}$ equation gives two $n$ : If $d < 0$ (or signs change across zero), partial sums can return to the same total; report both positive integer solutions unless the question forbids it.

Application snapshots (NCERT flavour)

Salary with annual increment $\to$ AP in $a_{n}$ .
Money-box deposits increasing by a fixed amount $\to$ $S_{n}$ .
Rows of plants decreasing by a fixed number $\to$ find $n$ from $a_{n} =$ last row count.
Simple interest year-end amounts forming an AP $\to$ $a_{n}$ for year $30$ , etc.

Exam-Oriented Analysis

Aspect	Detail
Weight	High; often 6–8 marks across term + sum + application
Question types	Is it AP?; $a_{n}$ ; find $n$ ; missing terms; $S_{n}$ ; word problems; multiples in an interval
Common mistakes	Using $a + n d$ instead of $a + (n - 1) d$ ; non-integer $n$ accepted; wrong $d$ sign when reversing
Scoring tips	Write formula; substitute with brackets; conclude “not a term” when $n \in / Z^{+}$

Question-type table

Style	Typical marks	Key check
Identify AP / find $d$	1–2	Several consecutive differences
Find $a_{n}$ or $n$	2–3	$(n - 1) d$ ; $n \in Z^{+}$
Missing terms in boxes	2	Use mid-term average properties / $a_{n}$
$S_{n}$ calculation	2–3	Correct formula form
Word problem (salary, rows, prizes)	3–4	Identify $a, d, n$ or $S_{n}$ from story

Common mistakes (expanded)

Writing $a_{n} = a + n d$ .
Taking $d =$ previous − next on a decreasing list.
Accepting $n = 7.5$ as a term index.
Using $\frac{n}{2} (a + l)$ with the wrong $n$ .
Forgetting both roots of the quadratic in $n$ for sum problems.

Scoring tips

Start with “Here $a = \dots$ , $d = \dots$ , $n = \dots$ ”.
Substitute into a boxed formula line.
If $n$ is not a positive integer, write explicitly: “Hence … is not a term of the AP.”

Method chooser

List / story
  → Is difference constant? If no, stop (not AP)
  → Need a particular term? Use a_n
  → Need total? Use S_n
  → Mixed S and a_n? Use a_n = S_n − S_{n−1}
  → From the end? Reverse or use l−(k−1)d