E ExamMaster

CBSE Class 10 · Mathematics

Arithmetic Progressions

Official NCERT chapter from Mathematics (book code jemh1). ExamMaster notes are original teaching at CBSE Class 10 depth.

This lesson follows the official chapter “Arithmetic Progressions” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 10
  • Medium level
  • 8 concepts

1Arithmetic Progressions

An arithmetic progression adds the same common difference d each step: 3, 7, 11, 15 has d = 4. The nth term is a + (n−1)d. A list whose gaps drift is not this AP. Two terms always have a difference; a whole list must keep it.

Calling 3, 7, 12 an AP with “almost 4” is a miss.

Figure. An AP adds the same d each step. Here d = 3, so 2, 5, 8, 11, 14.

How it works

  1. Subtract consecutive termsThe gaps.
  2. See if the gap staysAP if yes.
  3. Write a_n = a + (n−1)dThe rule.

2Sum of First n Terms of an AP

The sum of the first n terms is S_n = n/2 [2a + (n−1)d] (or n/2 (first + last) when the last is known). For a = 3, d = 4, n = 10: S_10 = 5 × (6 + 36) = 210. Pairing first with last is the why; missing the 1/2 is the miss.

Adding 10 × 3 and stopping is not S_10.

Figure. S_n for a = 10, d = 6 starts at the origin (n = 0) and reaches 306 at n = 9. The curve bends up because later terms are larger.

How it works

  1. Name a, d, nThe given.
  2. Write n/2 [2a+(n−1)d]The sum.
  3. For a check, last = a+(n−1)d and n/2 (first+last)The pair-form.

S_10 of 3, 7, 11, …

Sum the first 10 terms of the AP 3, 7, 11, …

  • a, d, n3, 4, 10
  • 2a+(n−1)d6+36=42
  • S_105×42=210

Pro tip. n/2 times that bracket; last term 39 also gives 5×(3+39).

3A definition is a test you can run

AP is a test: an ordered list plus a constant d. If you only have a pile, you have numbers, not a progression.

A pretty 3,7,11 banner is not the test until d is checked.

Figure. A number c sits in 2, 5, 8, 11, 14 only when 1 + (c − 2)/3 is a positive integer. 14 works; 13 does not.

How it works

  1. Write the ordered listThe object.
  2. Name d and show it staysThe test.
  3. Then the word has contentThe definition ran.

4Name the given before the unknown

The given is a, d (or two terms). The unknown is a_n or S_n. Copy d = 4 before you use 3 as if it were d.

Using n = 10 as a_n is a silent given.

Figure. Write a, d, and n first. Then a_n = a + (n − 1)d. Here 8 + 11 × 3 = 41.

How it works

  1. Copy a and dThe given.
  2. Name a_n or S_nThe unknown.
  3. Then plug nGiven first.

5One worked case is enough at this class

One AP-sum is enough. A page of the same 210 clone does not add a new d.

Ten twins of 3,7,11 are still one idea.

Figure. Worked AP a = 8, d = 7. The first five marks are 8, 15, 22, 29, 36; the ninth term is 64 and S9 is 324.

How it works

  1. Find one a_n or one S_nThe case.
  2. A second only as a checkOptional.
  3. StopThis class.

6A miss: swapping the school name for the picture

A staircase photo is a setting. The maths is a constant rise d. Steps that change height are not this AP.

A spiral stair is not d = 4.

Figure. The ninth term is 64; the sum of nine terms is 324. Swapping a_n for S_n is the miss — they are different asks.

How it works

  1. Keep the stair as a storyA setting.
  2. Write a and dThe maths.
  3. Then a_n or S_nDo not swap.

7Check by the opposite action or the opposite test

Check S_n by the other form n/2 (first+last), or check a_n by walking a few terms. 210 from both writings must match.

A walk that drifts d is not the opposite test for this AP.

Figure. Check a term by the opposite walk: 72 − 10 × 6 must rebuild a = 12. If it does not, the term was wrong.

How it works

  1. State 210 or a_10 = 39The claim.
  2. Rebuild with first+last or a walkThe check.
  3. Mend if they disagreeHonest.

8Keep the claim at this chapter, not the next

This chapter’s claim stops at AP terms and S_n. It does not steal triangle similarity, and it does not start a geometric-progression dump unless a later official idea names it.

A page that already “proves AAA” here misses the triangle chapter.

Figure. Equal added steps are this chapter. Doubling 3, 6, 12, 24 is a geometric list — leave it for a later chapter.

How it works

  1. Keep a, d, a_n, S_nThis chapter.
  2. Leave similar triangles for that chapterThe next map.
  3. Do not treat the map as already answeredClaim-size.
S_10 of 3, 7, 11, … is
  1. 210
  2. 30
  3. 39

5 × 42 = 210; 39 is the last term.

Notes

  • The official chapter title is “Arithmetic Progressions”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • a_n = a + (n−1)d
  • S_n = n/2 [2a+(n−1)d]

Recap

Hold these pegs from the official chapter “Arithmetic Progressions”. The wording is ExamMaster’s teaching, not a textbook recap.

Arithmetic Progressions
An arithmetic progression adds the same common difference d each step: 3, 7, 11, 15 has d = 4.
Sum of First n Terms of an AP
The sum of the first n terms is S_n = n/2 [2a + (n−1)d] (or n/2 (first + last) when the last is known).
A definition is a test you can run
AP is a test: an ordered list plus a constant d.
Name the given before the unknown
The given is a, d (or two terms). The unknown is a_n or S_n. Copy d = 4 before you use 3 as if it were d.
One worked case is enough at this class
One AP-sum is enough. A page of the same 210 clone does not add a new d.
A miss: swapping the school name for the picture
A staircase photo is a setting. The maths is a constant rise d. Steps that change height are not this AP.

Practise Arithmetic Progressions

Reading is free and needs no account. Practice, mocks and progress live in the app.

  • A 4-question practice set that ends the chapter
  • 1 quick check with worked explanations
  • Timed mocks scored with the real marking scheme
  • Readiness tracked per topic, kept on your device
Continue with Google — freeNo card, no trial. Works offline once installed.