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CBSE · Class 10 · Mathematics

Arithmetic Progressions 5.2: Arithmetic Progressions

5 concepts · 40 practice questions with answers · Free to read

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Common difference of an AP

From daily life

Steps on a staircase rise by the same height each time. A list that grows by the same amount each step is an arithmetic progression.

Simple definition

An arithmetic progression (AP) is a list of numbers in which each term is got by adding the same fixed number to the previous term. That fixed number is the common difference d = a₂ − a₁.

Formula: d = a₂ − a₁ = a₃ − a₂ = …

3+47+411+415+419

In the figure: 3, 7, 11, 15, 19 climbs by +4 each time, so d = 4.

Rule 1: Find d — Subtract any term from the next one.

  • Easy: 3, 7, 11, 15: d = 4.
  • Medium: 20, 17, 14, 11: d = −3 (decreasing).
  • Exam-type: 1/2, 1, 3/2, 2: d = 1/2.

Rule 2: Sign of d — d > 0 increases, d < 0 decreases, d = 0 is constant.

  • Easy: 5, 5, 5, 5: d = 0.
  • Medium: 10, 7, 4: d = −3 (decreasing AP).
  • Exam-type: −5, −2, 1, 4: d = 3.
Remember

The common difference can be negative, zero, a fraction or a decimal.

Common mistake

Subtracting in the wrong order: a₁ − a₂ instead of a₂ − a₁.

Memory clue

"Next minus previous."

Practice questions

Easy
  1. 1In an AP, the 4th term is 4 and the 6th term is 10. Find the common difference.
  2. 2In an AP, the 2nd term is 14 and the 3rd term is 19. Find the common difference.
  3. 3In an AP, the 1st term is −5 and the 3rd term is 11. Find the common difference.
Medium
  1. 4In an AP, the 2nd term is 10 and the 5th term is 16. Find the common difference.
  2. 5In an AP, the 5th term is 30 and the 7th term is 40. Find the common difference.
  3. 6In an AP, the 1st term is 15 and the 6th term is 55. Find the common difference.
Challenging
  1. 7In an AP, the 1st term is −2 and the 6th term is 8. Find the common difference.
  2. 8In an AP, the 5th term is −27 and the 7th term is −45. Find the common difference.
Show answers (try them yourself first)
  1. 13
  2. 25
  3. 38
  4. 42
  5. 55
  6. 68
  7. 72
  8. 8−9

Writing the terms of an AP

From daily life

Once you know where to start and the step size, you can write the whole list: add d again and again.

Simple definition

Given the first term a and the common difference d, the AP is a, a + d, a + 2d, a + 3d, ….

Formula: a, a + d, a + 2d, a + 3d, …

5+38+311+314+317

In the figure: a = 5, d = 3 gives 5, 8, 11, 14, 17.

Live figure

Writing the terms of an AP

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Rule 1: Write the first terms — Add d to each term to get the next.

  • Easy: a = 5, d = 3: 5, 8, 11, 14.
  • Medium: a = 10, d = −2: 10, 8, 6, 4.
  • Exam-type: a = −1, d = 0.5: −1, −0.5, 0, 0.5.

Rule 2: Continue the list — Find d from the given terms, then extend.

  • Easy: 2, 9, 16, … : d = 7, next 23, 30.
  • Medium: 50, 45, 40, … : d = −5, next 35, 30.
  • Exam-type: Each new term is the last term plus d.
Remember

The same d is added every time: the list never changes its step.

Common mistake

Multiplying by d instead of adding it.

Memory clue

"Start at a, keep adding d."

Practice questions

Easy
  1. 1In the usual notation, a stands for the ___ of an AP.
  2. 2An AP has first term 13 and common difference 2. What is its second term?
  3. 3An AP has first term −5 and common difference −5. What is its fifth term?
Medium
  1. 4An AP has first term −9 and common difference 3. What is its third term?
  2. 5An AP has first term −7 and common difference −5. What is its second term?
  3. 6The common difference of an AP is 3 and its term number 6 is 6. Find the first term.
Challenging
  1. 7The common difference of an AP is 5 and its term number 3 is 34. Find the first term.
  2. 8The common difference of an AP is 3 and its term number 3 is 15. Find the first term.
Show answers (try them yourself first)
  1. 1first term
  2. 215
  3. 3−25
  4. 4−3
  5. 5−12
  6. 6−9
  7. 724
  8. 89

Deciding whether a list is an AP

From daily life

To check a staircase is even, measure every step. A list is an AP only if every step has the same size.

Simple definition

A list is an AP if all the differences between consecutive terms are equal. If even one difference differs, it is not an AP.

Formula: a₂ − a₁ = a₃ − a₂ = a₄ − a₃

differences3, 3, 3equalAP, d = 3

In the figure: 2, 5, 8, 11 has differences 3, 3, 3, so it is an AP; 1, 4, 9, 16 has 3, 5, 7, so it is not.

Rule 1: AP or not? — Compute all consecutive differences.

  • Easy: 1, −1, −3, −5: differences −2, −2, −2: AP.
  • Medium: 1, 2, 4, 8: differences 1, 2, 4: not an AP.
  • Exam-type: 7, 7, 7, 7: differences 0: AP with d = 0.

Rule 2: Squares and other patterns — Patterns that multiply or square are not APs.

  • Easy: 1, 4, 9, 16 (squares): not an AP.
  • Medium: 3, 6, 12, 24 (doubling): not an AP.
  • Exam-type: 10, 20, 30, 40 (adding 10): an AP.
Remember

One unequal difference is enough to say "not an AP".

Common mistake

Checking only the first two differences and declaring it an AP.

Memory clue

"All steps equal or it is no AP."

Practice questions

Easy
  1. 1Is the list 4, 9, 16, 25, 36, … an arithmetic progression?
  2. 2Is the list 13, 21, 13, 21, 13, … an arithmetic progression?
  3. 3Is the list 1, 5, 1, 5, 1, … an arithmetic progression?
Medium
  1. 4Is the list 13, 19, 13, 19, 13, … an arithmetic progression?
  2. 5Is the list 9, 11, 14, 18, 24, … an arithmetic progression?
  3. 6Is −14, −17, −20, −24 an arithmetic progression?
Challenging
  1. 7Is 22, 28, 34, 38 an arithmetic progression?
  2. 8Is −10, −5, 0, 4 an arithmetic progression?
Show answers (try them yourself first)
  1. 1No
  2. 2No
  3. 3No
  4. 4No
  5. 5No
  6. 6No
  7. 7No
  8. 8No

Unknown value so that the terms are in AP

From daily life

If three numbers are evenly spaced, the middle one is exactly halfway between the other two.

Simple definition

If a, b, c are in AP then b − a = c − b, i.e. 2b = a + c. Use this to find an unknown such as k.

Formula: 2b = a + c

k, 2k+1, 4k−12(2k+1) = k + 4k − 1k = 3

In the figure: k, 2k + 1, 4k − 1 in AP: 2(2k + 1) = k + 4k − 1, so 4k + 2 = 5k − 1, k = 3: terms 3, 7, 11.

Rule 1: Use 2b = a + c — Write the equation and solve for the unknown.

  • Easy: x, 10, 16 in AP: 20 = x + 16, x = 4.
  • Medium: k + 2, 4k − 6, 3k − 2: 8k − 12 = 4k, k = 3: terms 5, 6, 7.
  • Exam-type: Check: 7 − 3 = 4 and 11 − 7 = 4 ✓.

Rule 2: Using equal differences — Equate two differences when the middle term form is awkward.

  • Easy: p, 7, 13: 7 − p = 13 − 7 = 6, p = 1.
  • Medium: 5, m, 15: m − 5 = 15 − m, m = 10.
  • Exam-type: m is the average of its neighbours.
Remember

The middle term of three terms in AP is the average of the other two.

Common mistake

Writing 2b = a − c or forgetting to put brackets around 2k + 1.

Memory clue

"Twice the middle equals the sum of the ends."

Practice questions

Easy
  1. 1If −8, x, −20 are in AP, find x.
  2. 2If 5, x, −13 are in AP, find x.
  3. 3If 9, x, 11 are in AP, find x.
Medium
  1. 4The numbers −1, −5, y, x, −17 are in AP. Find the value of x − y.
  2. 5For what value of k are 5k − 4, −k − 1 and −2k + 7 three consecutive terms of an AP?
  3. 6For what value of k are −k + 9, −k and 2k three consecutive terms of an AP?
Challenging
  1. 7For what value of k are −k − 11, 4k + 8 and −3k − 9 three consecutive terms of an AP?
  2. 8For what value of k are 2k − 7, −k + 8 and 4k − 25 three consecutive terms of an AP?
Show answers (try them yourself first)
  1. 1−14
  2. 2−4
  3. 310
  4. 4−4
  5. 5−1
  6. 6−3
  7. 7−3
  8. 86

Situations that form an AP

From daily life

Taxi fares, savings and seating rows grow by a fixed amount, so they form APs. Doubling and interest do not.

Simple definition

A situation gives an AP when each step adds (or subtracts) the same amount. List a few values, find the differences, and check they are equal.

taxi15, 23, 31savings100, 150, 200doubling1, 2, 4 (not AP)

In the figure: Taxi: ₹15 for the first km and ₹8 for each extra km: 15, 23, 31, … is an AP with d = 8.

Rule 1: AP situations — The fixed amount added each time is d.

  • Easy: Seats in rows 15, 18, 21, …: d = 3.
  • Medium: Savings ₹100, 150, 200, …: d = 50.
  • Exam-type: Ladder rungs 45, 43, 41, … cm: d = −2.

Rule 2: Not AP situations — Multiplying each time or percentage growth does not give an AP.

  • Easy: Cells 1, 2, 4, 8 (splitting): not an AP.
  • Medium: Compound interest amounts: not an AP.
  • Exam-type: Simple interest amounts each year: an AP.
Remember

If the same amount is added each time, it is an AP; if the same factor is multiplied, it is not.

Common mistake

Calling any increasing list an AP; the increase must be the same each step.

Memory clue

"Add the same: AP. Multiply the same: not AP."

Practice questions

Easy
  1. 1A student saves ₹34 in week 1 and increases the saving by ₹11 every week. Do the weekly savings ₹34, ₹45, ₹56, … form an AP?
  2. 2The amounts at the end of 1, 2 and 3 years on ₹1000 at 20% compound interest are about ₹1200, ₹1440, ₹1728. Do they form an AP?
  3. 3A student saves ₹7 in week 1 and increases the saving by ₹6 every week. Do the weekly savings ₹7, ₹13, ₹19, … form an AP?
Medium
  1. 4A student saves ₹21 in week 1 and increases the saving by ₹14 every week. Do the weekly savings ₹21, ₹35, ₹49, … form an AP?
  2. 5A student saves ₹40 in week 1 and increases the saving by ₹13 every week. Do the weekly savings ₹40, ₹53, ₹66, … form an AP?
  3. 6A taxi charges ₹22 for the first km and ₹16 for each additional km. Using the AP of fares for 1 km, 2 km, 3 km, …, find the fare for 14 km (in ₹).
Challenging
  1. 7A taxi charges ₹37 for the first km and ₹16 for each additional km. Using the AP of fares for 1 km, 2 km, 3 km, …, find the fare for 7 km (in ₹).
  2. 8A taxi charges ₹46 for the first km and ₹14 for each additional km. Using the AP of fares for 1 km, 2 km, 3 km, …, find the fare for 13 km (in ₹).
Show answers (try them yourself first)
  1. 1Yes
  2. 2No
  3. 3Yes
  4. 4Yes
  5. 5Yes
  6. 6230
  7. 7133
  8. 8214