Is it correct to say: Using aₙ = a + n·d (missing the '−1')?

Correct: aₙ = a + (n − 1)d. The 1st term is 'a', so only n−1 steps to reach the nth.

Is it correct to say: Assuming any three numbers with a difference form an AP?

Difference must be CONSTANT. 2, 4, 8 is NOT an AP (differences 2 and 4).

Is it correct to say: Mixing up Sₙ formulas — using (a + l) form when l is unknown?

Use Sₙ = n/2 (a + l) only when l is given. Otherwise use Sₙ = n/2 [2a + (n−1)d].

Is it correct to say: Getting non-integer n in 'which term' problems?

n must be a positive integer. Non-integer means the value is NOT a term of the AP.

Is it correct to say: Forgetting to reject negative n from the quadratic in word problems?

In loan / instalment / age problems n > 0 by nature. Discard negative root.

Is it correct to say: Confusing common difference d with the ratio r of a GP?

AP → constant DIFFERENCE (add d). GP → constant RATIO (multiply by r). Class 10 syllabus deals with AP only.

Arithmetic Progressions

General term, sum of first n terms, applications of arithmetic progressions in word problems — NCERT Class 10 Maths Ch 5

📐 8 formulas✏️ 3 examples🎯 6 practice⚖️ 5 marks🏫 CBSE📚 Class 10✓ 2025–26 syllabus

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Board Exam Tips

→aₙ = a + (n−1)d and Sₙ = n/2 [2a + (n−1)d] MUST be memorised.
→For last-term problems, use Sₙ = n/2 (a + l) where l is the last term.
→'Which term is …' → equate aₙ = value and solve for n.
→Word problems (savings, loan instalments, wages) — set up 'a' and 'd' before touching the formula.
→Middle-term property: a, b, c in AP ⇔ 2b = a + c. Handy for proofs.

📐 Formulas(8)

General Term (nth term)★ Board fav

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

Symbol	Meaning
$a$	First term of the AP
$d$	Common difference (a₂ − a₁ = a₃ − a₂)
$n$	Position (natural number)

Common Difference

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

Sum of First n Terms★ Board fav

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

Sum when Last Term Is Known★ Board fav

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

Middle-term (Three in AP)

2 b = a + c

n-th Term from End

a_{n}^{'} = l - (n - 1) d

Number of Terms n from Given aₙ

n = \frac{a _{n} - a}{d} + 1

Standard Sums

\sum_{k = 1}^{n} k = \frac{n ( n + 1 )}{2}, \sum_{k = 1}^{n} (2 k - 1) = n^{2}

✏️ Solved Examples

1Solved Exampleeasy3 steps

Find the 20th term of the AP: 3, 8, 13, 18, …

Identify first term and common difference

a = 3, d = 8 - 3 = 5

2Solved Exampleboard4 steps

Find the sum of the first 30 terms of the AP: 8, 3, −2, −7, …

Identify a and d

a = 8, d = 3 - 8 = - 5

3Solved ExampleHOTS6 steps

A man repays a loan of ₹3250 by paying ₹20 in the first month and then increasing the payment by ₹15 every month. How many months will he take to clear the loan?

Monthly payments form an AP: 20, 35, 50, … with a = 20, d = 15.

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Using aₙ = a + n·d (missing the '−1')
✓Correct: aₙ = a + (n − 1)d. The 1st term is 'a', so only n−1 steps to reach the nth.
2
Assuming any three numbers with a difference form an AP
✓Difference must be CONSTANT. 2, 4, 8 is NOT an AP (differences 2 and 4).
3
Mixing up Sₙ formulas — using (a + l) form when l is unknown
✓Use Sₙ = n/2 (a + l) only when l is given. Otherwise use Sₙ = n/2 [2a + (n−1)d].
4
Getting non-integer n in 'which term' problems
✓n must be a positive integer. Non-integer means the value is NOT a term of the AP.
5
Forgetting to reject negative n from the quadratic in word problems
✓In loan / instalment / age problems n > 0 by nature. Discard negative root.
6
Confusing common difference d with the ratio r of a GP
✓AP → constant DIFFERENCE (add d). GP → constant RATIO (multiply by r). Class 10 syllabus deals with AP only.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Find the 25th term of the AP 5, 8, 11, 14, …
Q2
How many terms of the AP 21, 18, 15, … make a sum of 78?
Q3
Find the sum: 1 + 3 + 5 + … + 199.
Q4
If the 7th term of an AP is 34 and the 13th term is 64, find the AP.
Q5
Which term of the AP 3, 8, 13, 18, … is 78?
Q6
The sum of first 8 multiples of 3 is …

📝 Notes

Arithmetic Progressions

An arithmetic progression (AP) is a sequence in which each term after the first differs from the previous by a constant, called the common difference (d).

Example: 5, 8, 11, 14, 17, … (a = 5, d = 3)

Two master formulas

Almost every AP question in the Class 10 board yields to one of these:

nth term: aₙ = a + (n − 1)d
Sum of n terms: Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)

Everyday examples of AP

Money saved monthly with a fixed increment.
Rows of seats in a theatre (each row has same extra seats).
Rolls of a stack — each roll placed atop with a fixed offset.
Simple interest amounts each year for a fixed principal.

Middle-term property

Three numbers a, b, c are in AP iff 2b = a + c. Use this for algebraic proofs.

Choosing convenient forms

3 numbers in AP → let them be (a − d), a, (a + d). Symmetric ⇒ simplifies algebra.
4 numbers in AP → (a − 3d), (a − d), (a + d), (a + 3d).
5 numbers in AP → (a − 2d), (a − d), a, (a + d), (a + 2d).