🔢 Class 10 Mathematics — Arithmetic Progressions
Interactive Visual Revision Notes
Aligned with the Class 10 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.
🌎 Real World Connection: Imagine saving ₹200 in January, ₹400 in February, ₹600 in March — you add ₹200 every month. This is an arithmetic progression! Knowing the pattern lets you instantly calculate your total savings after any number of months without listing every month. Bus schedules, stair steps, and salary increments all follow A.P. patterns.
A list of numbers with a fixed gap between them.
We call this gap the common difference (d).
Example: 3, 7, 11, 15, 19 ... (d = 4)
Each number is called a term.
Short form: A.P.
🔢 Topic 2: The Common Difference
d = next term − previous term.
d must be the same for every pair.
d can be positive (increasing A.P.).
d can be negative (decreasing A.P.).
d can be zero (constant A.P.).
🔢 Topic 3: General Term Formula
The n-th term is written as aₙ.
Formula: aₙ = a + (n − 1)d
Here a = first term.
And d = common difference.
This finds any term without listing all.
🔢 Topic 4: Finding n (Which Term?)
If a term value is given, find its position.
Use: n = (aₙ − a) ÷ d + 1
Example: In A.P. 5, 9, 13 ... is 41 a term?
n = (41 − 5) ÷ 4 + 1 = 36 ÷ 4 + 1 = 10
Yes! 41 is the 10th term.
🔢 Topic 5: Sum of First n Terms
Written as Sₙ.
Formula 1: Sₙ = n/2 × (2a + (n−1)d)
Formula 2: Sₙ = n/2 × (a + l)
Use Formula 2 when the last term (l) is known.
Both formulas give the same answer.
🔢 Topic 6: Useful Quick Formulas
Sum of first n naturals: n(n+1)/2
Sum of first n odds: n²
Sum of first n evens: n(n+1)
These are all special A.P. results.
They save time in many questions.
🔢 Topic 7: Finding d When Two Terms Are Known
If the p-th and q-th terms are given:
d = (a_q − a_p) ÷ (q − p)
Example: a₃ = 11 and a₇ = 23
d = (23 − 11) ÷ (7 − 3) = 12 ÷ 4 = 3
Then a = a₃ − 2d = 11 − 6 = 5
🔢 Topic 8: Checking If a Sequence Is an A.P.
Find the difference between every pair.
If all differences are equal, it is an A.P.
Check: 4, 9, 14, 19 → d = 5, 5, 5 ✔
Check: 2, 4, 7, 11 → d = 2, 3, 4 ✘
Even one unequal gap means not an A.P.
🔢 Topic 9: Arithmetic Mean
The arithmetic mean of a and b is (a+b)/2.
It is the middle value between two numbers.
In an A.P., each term is the AM of its neighbours.
Example: AM of 8 and 20 = (8+20)/2 = 14
So 8, 14, 20 form an A.P.
🔢 Topic 10: Real-Life A.P. Patterns
Saving money: save ₹10 more each week.
Stacking bricks: each row has 2 fewer.
Taxi fare: fixed start + per km charge.
Staircase steps: equal height increase.
A.P. appears everywhere in daily life.
🎨 Section 2 — Easy Illustrations
Picture 1 — Each term increases by the same amount
Picture 2 — How to find any term in an A.P.
Picture 3 — Two ways to find the sum
Picture 4 — A.P. can go up, come down, or stay flat
Picture 5 — Saving ₹5 more each week is an A.P.
✏️ Section 3 — Worked Examples
✏️ Example 1: Find the 15th Term
A.P. is 7, 13, 19, 25, ... Find the 15th term.
1 First term: a = 7
2 Common difference: d = 13 − 7 = 6
3 Formula: aₙ = a + (n − 1)d
4 a₁₅ = 7 + (15 − 1) × 6
5 a₁₅ = 7 + 14 × 6 = 7 + 84 = 91
🎯 The 15th term = 91
✏️ Example 2: Find the Sum of First 20 Terms
A.P. is 4, 11, 18, 25, ... Find S₂₀.
1 a = 4, d = 11 − 4 = 7
2 Formula: Sₙ = n/2 × [2a + (n−1)d]
3 S₂₀ = 20/2 × [2(4) + (20−1)(7)]
4 S₂₀ = 10 × [8 + 133]
5 S₂₀ = 10 × 141 = 1410
🎯 Sum of first 20 terms = 1410
✏️ Example 3: Which Term Is 99?
In the A.P. 3, 9, 15, 21, ... which term is 99?
1 a = 3, d = 9 − 3 = 6
2 We need: aₙ = 99
3 99 = 3 + (n − 1) × 6
4 96 = (n − 1) × 6
5 n − 1 = 16, so n = 17
🎯 99 is the 17th term
📊 Section 4 — Cheat Sheet
Concept
Formula
Remember
Common difference
d = a₂ − a₁
Same gap everywhere
n-th term
aₙ = a + (n−1)d
Find any term directly
Sum (no last term)
Sₙ = n/2 × [2a + (n−1)d]
Use when l is unknown
Sum (last term known)
Sₙ = n/2 × (a + l)
Shorter and easier
Which term?
n = (aₙ − a)/d + 1
Find position of a value
Arithmetic Mean
AM = (a + b)/2
Middle of two numbers
Sum of n naturals
n(n+1)/2
1+2+3+...+n
Sum of n odds
n²
1+3+5+...+(2n−1)
d > 0
A.P. increases
Going up
d < 0
A.P. decreases
Going down
📄 Section 5 — Take-Home Handout
✍️ A. True or False
1. In an A.P., every difference must be equal. ___
2. d can be negative in an A.P. ___
3. The 1st term of any A.P. is always positive. ___
4. Sₙ = n/2 × (a + l) works only when d > 0. ___
5. The AM of 10 and 30 is 20. ___
✍️ B. Fill in the Blanks
1. aₙ = a + ________
2. Sum of first 10 naturals = ________
3. If d = 0, all terms are ________
4. In 8, 14, 20, 26 the value of d = ________
5. Sum of first 5 odd numbers = ________
✍️ C. Solve These
1. Find the 12th term of the A.P.: 2, 6, 10, 14, ...
Show your work here
2. Find the sum of first 8 terms of: 5, 10, 15, 20, ...
Show your work here
🔗 D. Match the Columns
Column A
Column B
1. aₙ = a + (n−1)d
a. Sum formula
2. Sₙ = n/2(a+l)
b. n-th term
3. d < 0
c. Constant A.P.
4. d = 0
d. Decreasing A.P.
🎨 E. Write the First 5 Terms
Write the first 5 terms of the A.P. where a = −3 and d = 4.
Show your work here
✅ Detailed Solutions
A. True or False — Answers
1TRUE — In every A.P. the difference between consecutive terms must be equal.
2TRUE — d can be negative (decreasing A.P.), e.g. 10, 7, 4, 1 ... (d = −3).
3FALSE — The 1st term a can be negative, e.g. −5, −2, 1, 4 ... (a = −5).
4FALSE — Sₙ = n/2 (a + l) works for any A.P. regardless of whether d > 0.
5TRUE — AM of 10 and 30 = (10 + 30)/2 = 20.
B. Fill in the Blanks — Answers
1 aₙ = a + (n − 1)d
2 Sum of first 10 naturals = 10 × 11 / 2 = 55
3 If d = 0, all terms are equal
4 In 8, 14, 20, 26 the value of d = 14 − 8 = 6
5 Sum of first 5 odd numbers = 5² = 25
C1. Find the 12th term of 2, 6, 10, 14, ...
1 First term a = 2, common difference d = 6 − 2 = 4