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Arithmetic Progressions, Finally in Real Life

This chapter is built to feel natural, not memorised. We connect A.P. to railway berths, auto fares, battery drop, wedding shagun, rangoli patterns, savings habits, and exam questions that students actually face in Class 10.

Class 10 CBSE / ICSEPractical Indian examplesNth term and sum made clearExam-focused revision

Chapter Roadmap

01Sequence and Arithmetic Progression 02Common Difference 03General Form of an A.P.04Nth Term Formula 05Sum of the First n Terms 06Arithmetic Mean 07Word Problems and Exam Strategy

General form

a, a+d, a+2d, \dots

Difference rule

$d=$ next term $-$ previous term

Nth term

$a_n=a+(n-1)d$

Total sum

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

What Is a Sequence and What Makes It an Arithmetic Progression?

A sequence is an ordered list; an A.P. is a list that moves with equal jumps.

A sequence is just numbers written in a definite order. An arithmetic progression, or A.P., is a special sequence in which each new term is obtained by adding or subtracting the same fixed number.

So the real test is simple: if the gap between consecutive terms stays constant, the sequence is an A.P. If the gap keeps changing, it is not.

Key Formula

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

Here

a

is the first term and

d

is the common difference.

Everyday Indian Example

Railway Side Lower Berths

Side Lower berth numbers in many Indian train coaches often appear as $7,15,23,31,\dots$ .

Each new Side Lower berth is exactly 8 more than the previous one, so the pattern has a fixed jump.

That means this is an A.P. with $a=7$ and $d=8$ .

Halwai tray pattern

The rows

1,3,5,7,\dots

increase by 2 each time, so this is an A.P. with

d=2

Phone battery drop

The battery levels

100,97,94,91,\dots

decrease by 3 each time, so this is also an A.P., but with negative difference.

Square numbers

The pattern

1,4,9,16,\dots

is not an A.P. because the differences are

3,5,7,\dots

, not constant.

Solved Example

Check whether a sequence is an A.P.

Check whether

1,4,7,10,13,\dots

is an arithmetic progression.

Show Solution

Find the consecutive differences.

$4-1=3$ , $7-4=3$ , $10-7=3$ , and $13-10=3$ .

All the differences are equal, so the sequence is an arithmetic progression.

Here $a=1$ and $d=3$ .

Practice

State whether each is an A.P. and write the common difference if it exists:

12,7,2,-3,-8,\dots

2,5,10,17,\dots

, and

0.6,1.7,2.8,3.9,\dots

Identifying the Common Difference

The common difference is the number you keep adding or subtracting every time.

To find the common difference, subtract a term from the next term. Never guess a pattern by just staring at the first and last terms.

A positive value of

d

means the sequence is increasing. A negative value means it is decreasing. Both are perfectly valid arithmetic progressions.

Teacher Insight

The safest habit is to check at least three consecutive differences. A pattern only counts as an A.P. when every jump stays the same.

Everyday Indian Example

Battery While Streaming Highlights

Suppose your battery percentages go $100,97,94,91,\dots$ while watching a long match.

Each time the battery drops by 3 percentage points, so $d=-3$ .

This shows that an A.P. can model decline as naturally as growth.

Wedding band volume

If the speaker levels go

10,12,14,16,\dots

, then the common difference is

d=2

Water tank levels

If the water level falls as

80,75,70,65,\dots

, then the common difference is

d=-5

Daily step target

If Riya walks

3000,3500,4000,4500,\dots

steps, the common difference is

d=500

Solved Example

Find the common difference

Find the common difference of

-9,-7,-5,-3,\dots

Show Solution

Compute consecutive differences:

$-7-(-9)=2$ , $-5-(-7)=2$ , and $-3-(-5)=2$ .

So the common difference is $d=2$ .

Practice

Write the common difference for

5,8,11,14,\dots

40,35,30,25,\dots

, and

-1.25,-1.50,-1.75,-2.00,\dots

Writing the General Form of an A.P.

Once you know the first term and the step size, the whole sequence is easy to build.

If the first term is

a

and the common difference is

d

, then the second term is

a+d

, the third term is

a+2d

, and so on.

This general form is useful because every later formula of the chapter comes from this same pattern.

Mental Shortcut

If $d$ is positive, the A.P. climbs upward like a staircase. If $d$ is negative, it comes down like a countdown timer or a draining battery.

Key Formula

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

Everyday Indian Example

Auto Rickshaw Fare Pattern

Suppose the fare is Rs. 30 for the first kilometre and rises by Rs. 15 for every extra kilometre.

Then the fare pattern becomes Rs. $30,45,60,75,\dots$ .

So this is an A.P. with $a=30$ and $d=15$ .

Solved Example

Write an A.P. from a and d

Write the A.P. whose first term is

10

and common difference is

3

Show Solution

Start with 10 and keep adding 3.

So the sequence is $10,13,16,19,22,\dots$ .

Practice

Write the first five terms of the A.P. if (i)

a=4

d=-3

(ii)

a=\frac13

d=\frac43

(iii)

a=-1.25

d=-0.25

Finding the Nth Term

The nth-term formula lets you jump directly to any required term.

To go from the first term to the second term, you add

d

once. To reach the third term, you add

d

twice. To reach the

n

th term, you add

d

exactly

(n-1)

times.

That is why the formula for the

n

th term is not

a+nd

but

a+(n-1)d

Common Mistake

Students often use $nd$ instead of $(n-1)d$ . Remember: the first term is already counted in $a$ , so you only count the jumps after it.

Key Formula

a_n=a+(n-1)d

Everyday Indian Example

Auto Meter for a 12 km Ride

For the fare pattern Rs. $30,45,60,75,\dots$ , we have $a=30$ and $d=15$ .

The fare for the 12th step is:

a_{12}=30+(12-1)\times15=30+165=195

So the 12th amount is Rs. 195.

Cricket singles pattern

If a batter moves like

8,9,10,11,\dots

after taking one single each ball, that pattern is an A.P. with

a=8

and

d=1

Salary revision pattern

A monthly allowance such as Rs. 500, Rs. 600, Rs. 700, \dots is also an A.P., and the nth payment can be found directly using

a_n

Solved Example

Find a specific term

Find the 15th term of the A.P.

7,11,15,19,\dots

Show Solution

Here $a=7$ , $d=4$ , and $n=15$ .

a_{15}=7+(15-1)\times4=7+56=63

So the 15th term is 63.

Practice

Find (i) the 20th term of

5,8,11,14,\dots

and (ii) the 15th amount if a pattern starts at Rs. 500 and increases by Rs. 100 each time.

Finding the Sum of the First n Terms

Use this whenever the question asks for the total, not just one term.

If a question asks for "total," "altogether," or "how much in all," then it is usually a sum problem, not an nth-term problem.

The sum formula saves time because you do not have to add every term individually.

Key Formula

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

S_n=\frac{n}{2}(a+l)

Use this form when the last term

l

is known.

Everyday Indian Example

Diwali Rangoli with Marigolds

Suppose the innermost circle uses 12 flowers and every next circle uses 6 more flowers.

Then the numbers of flowers are $12,18,24,\dots$ with $a=12$ , $d=6$ .

For 10 circles:

S_{10}=\frac{10}{2}[2(12)+9(6)]=5(24+54)=390

So you need 390 flowers in total.

Wedding shagun

If cousins receive Rs. 500, Rs. 600, Rs. 700, \dots, then the total amount distributed is found using

S_n

, not

a_n

Daily savings

If a shopkeeper saves Rs. 50, Rs. 60, Rs. 70, \dots for 30 days, the total savings come from the sum formula.

Bulb decoration

Rows using

8,12,16,20,\dots

bulbs are a classic A.P. total-count problem.

Solved Example

Find a total

Find the sum of the first 20 terms of

3,7,11,15,\dots

Show Solution

Here $a=3$ , $d=4$ , and $n=20$ .

S_{20}=\frac{20}{2}[2(3)+19(4)]

=10[6+76]=10\times82=820

So the required sum is 820.

Practice

1. A tuition teacher earns Rs. 25, Rs. 30, Rs. 35, Rs. 40, \dots in a pattern. Find the total in the first 12 days.
2. A festive light decoration uses

8,12,16,20,\dots

bulbs in each row. How many bulbs are needed for the first 15 rows?

Arithmetic Mean

The arithmetic mean is the exact balancing number between two terms.

If three numbers are in A.P., then the middle number is the arithmetic mean of the other two.

It is exactly halfway between them, so it is found by taking their average.

Visual Meaning

The arithmetic mean is simply the number that sits in the centre so that the gap on both sides is equal.

Key Formula

\text{Arithmetic Mean}=\frac{a+b}{2}

Everyday Indian Example

Dhaba on the Highway

Suppose you cross a toll plaza at kilometre 20 and your destination is at kilometre 100.

The perfectly halfway dhaba point is:

\frac{20+100}{2}=60

So the ideal stop is near the 60 km mark.

Solved Example

Find an arithmetic mean

Find the arithmetic mean between 14 and 30.

Show Solution

Use the average formula:

\frac{14+30}{2}=\frac{44}{2}=22

So the arithmetic mean is 22, and $14,22,30$ form an A.P.

Practice

Find (i) the arithmetic mean of 35 and 79, and (ii) one arithmetic mean between 8 and 18.

Word Problems and Exam Strategy

Translate the story into a, d, n first, then choose the correct formula.

Most A.P. word problems become easy when you first write the pattern clearly and identify

a

d

, and what the question is asking for.

If the question wants one specific term, use

a_n

. If it wants the total, use

S_n

. If it wants the middle balancing number, use the arithmetic mean idea.

Three-Step Strategy

Write the pattern in order.
Identify $a$ , $d$ , and whether the question asks for a term or a total.
Use $a_n$ for one term, $S_n$ for the total, and arithmetic mean for a balanced middle value.

Question says '15th day'

That is one specific term, so think of the nth-term formula.

Question says 'in 15 days altogether'

That means a total, so think of the sum formula.

Question says 'middle value'

That is an arithmetic mean situation.

Solved Example

Mixed exam-style problem

A music teacher charges Rs. 200 for the first class and increases the fee by Rs. 20 for each next session. Find the fee for the 18th session and the total fee for 18 sessions.

Show Solution

Here $a=200$ , $d=20$ , and $n=18$ .

Fee for the 18th session:

a_{18}=200+(18-1)\times20=200+340=540

Total fee for 18 sessions:

S_{18}=\frac{18}{2}[2(200)+17(20)]

=9(400+340)=9\times740=6660

So the 18th fee is Rs. 540 and the total is Rs. 6660.

Practice

1. A food vlogger saves Rs. 100, Rs. 150, Rs. 200, Rs. 250, \dots every month. Find the amount saved in the 18th month.
2. In a stadium, one row has 22 seats, the next has 26, the next has 30, and so on. Find the number of seats in the 25th row.
3. A child makes a rangoli with

9,13,17,21,\dots

petals in successive layers. Find the total petals in the first 14 layers.
4. Find the arithmetic mean between 126 and 194.

Quick Revision

Concept	Key Idea
General form	$a, a+d, a+2d, \dots$
Common difference	$d=$ next term $-$ previous term
Nth term	$a_n=a+(n-1)d$
Sum of first n terms	$S_n=\dfrac{n}{2}[2a+(n-1)d]$
Alternative sum form	$S_n=\dfrac{n}{2}(a+l)$
Arithmetic mean	$\dfrac{a+b}{2}$
Exam rule	One term means $a_n$ ; total means $S_n$ .

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