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Arithmetic Progressions – Class 10 Mathematics notes, important questions and MCQs

NCERT Class 10 Mathematics, chapter 5. Notes, key terms, important questions with answers and MCQs.

Chapter overview

An AP is a list of numbers where each term is the previous term plus a fixed common difference d. We use formulas for the nth term and the sum of n terms.

Basics

General form: a, a + d, a + 2d, …
d = a₂ − a₁ = a₃ − a₂ (must be the same throughout).
d can be positive, negative or zero.

nth term

aₙ = a + (n − 1)d.
nth term from the end = l − (n − 1)d, where l is the last term.
Three numbers in AP: take a − d, a, a + d.

Sum of n terms

Sₙ = n/2 [2a + (n − 1)d].
Sₙ = n/2 (a + l) when the last term is known.
aₙ = Sₙ − Sₙ₋₁.

Key terms

Common difference: the fixed amount added to get the next term
Arithmetic progression: a sequence with a constant common difference
nth term: aₙ = a + (n − 1)d, the general term of an AP

Formulas

aₙ = a + (n − 1)d
Sₙ = n/2 [2a + (n − 1)d]
Sₙ = n/2 (a + l)
Sum of first n natural numbers = n(n + 1)/2

Important questions with answers

Find the 10th term of the AP 2, 7, 12, …2 marks

a = 2, d = 5. a₁₀ = 2 + 9 × 5 = 47.

Which term of the AP 3, 8, 13, 18, … is 78?3 marks

a = 3, d = 5. 3 + (n − 1)5 = 78, so (n − 1)5 = 75, n − 1 = 15, n = 16. The 16th term is 78.

Find the sum of the first 40 positive integers divisible by 6. Also find how many terms of 24, 21, 18, … give a sum of 78.5 marks

First part: 6, 12, …, 240 is an AP with a = 6, d = 6, n = 40. S₄₀ = 40/2 × (6 + 240) = 20 × 246 = 4920. Second part: a = 24, d = −3. Sₙ = n/2[48 − 3(n − 1)] = 78 gives n(51 − 3n) = 156, i.e. 3n² − 51n + 156 = 0, n² − 17n + 52 = 0, (n − 4)(n − 13) = 0. So n = 4 or 13; both work because terms after the 9th are negative and cancel out.

Quick revision

aₙ = a + (n − 1)d.
Sₙ = n/2 [2a + (n − 1)d].
Take a − d, a, a + d for three terms.

Exam tips

Check d between more than one pair of terms before calling it an AP.
n must be a positive integer — reject fractional or negative n.

Practice MCQs

The common difference of 3, 7, 11, 15, … is
1. 4
2. 3
3. 7
4. −4
Show answer
Answer: (a) 4. 7 − 3 = 4.
The 10th term of 2, 5, 8, … is
1. 30
2. 29
3. 32
4. 27
Show answer
Answer: (b) 29. 2 + 9 × 3 = 29.
Sum of the first 20 natural numbers is
1. 210
2. 200
3. 220
4. 190
Show answer
Answer: (a) 210. 20 × 21/2 = 210.
Which term of 3, 8, 13, … is 78?
1. 16th
2. 15th
3. 17th
4. 14th
Show answer
Answer: (a) 16th. 3 + (n − 1)5 = 78 ⇒ n = 16.
Sum of the first 15 terms of 2, 6, 10, … is
1. 390
2. 420
3. 480
4. 450
Show answer
Answer: (d) 450. 15/2 [4 + 14 × 4] = 15/2 × 60 = 450.
If 2k + 1, 13 and 5k − 3 are in AP, k =
1. 3
2. 4
3. 5
4. 2
Show answer
Answer: (b) 4. 2 × 13 = 2k + 1 + 5k − 3 ⇒ 7k = 28 ⇒ k = 4.
How many two-digit numbers are divisible by 3?
1. 31
2. 29
3. 30
4. 33
Show answer
Answer: (c) 30. 12, 15, …, 99: (99 − 12)/3 + 1 = 30.
If Sₙ = 3n² + 2n, the common difference is
1. 3
2. 5
3. 6
4. 8
Show answer
Answer: (c) 6. a₁ = S₁ = 5, a₂ = S₂ − S₁ = 16 − 5 = 11, d = 6.

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Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.