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CBSE Class 9 · Mathematics

Predicting What Comes Next: Exploring Sequences and Progressions

Official NCERT chapter from Ganita Manjari Part I (book code iemh1). ExamMaster notes are original teaching at CBSE Class 9 depth.

This lesson follows the official chapter “Predicting What Comes Next: Exploring Sequences and Progressions” in Ganita Manjari Part I. 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 9
  • Medium level
  • 8 concepts

1Introduction to Sequences

A sequence is an ordered list: first term, second term, … The order is the object. {3, 7, 11} as a set forgets which came first; 3, 7, 11, … as a sequence keeps the next.

A bag of numbers with no next-arrow is not this list.

Figure. A sequence is an ordered list. 2, 5, 8, 11 sits on a line with a steady +3 hop. The next mark, if the same hop holds, would be 14.

How it works

  1. Write a first terma1.
  2. Write what comes next as a rule or a listThe sequence.
  3. Refuse a unordered pile as the same objectOrder matters.

2Explicit Rule for a Sequence

An explicit rule names the nth term from n alone: a_n = 3 + (n−1)×4 gives 3, 7, 11, 15, … You do not need the previous term once n is known. A rule that still asks for a_{n−1} is not this explicit form.

A pretty closed formula that fails n = 1 is not finished.

Figure. An explicit rule names a_n from n alone: a_n = 2n + 1. The dots (1, 3), (2, 5), (3, 7), (4, 9), (5, 11) lie on one rising line. You do not need the previous term.

How it works

  1. Write n = 1, 2, 3The index.
  2. Plug into the formulaThe terms.
  3. Check the first term matches the listExplicit earned a test.

a_n = 3 + (n−1)×4

Find the 5th term of 3, 7, 11, 15, … using an explicit rule.

  • Rulea_n = 3 + (n−1)×4
  • n = 53 + 16
  • a519

Pro tip. Explicit means n in, term out — no walk through 2, 3, 4 unless you want a check.

3Recursive Rule for a Sequence

A recursive rule names the next from the previous: a1 = 3, a_{n} = a_{n−1} + 4. You must walk. Recursion is a machine with a start and a step; a formula with only n and no start is the other heading.

Writing “add 4” with no first term does not start the machine.

Figure. A recursive rule needs a start and a hop: a₁ = 4 and aₙ = aₙ₋₁ + 3. Each term is born from the one before, so 4, 7, 10, 13.

How it works

  1. Name a1The start.
  2. Name how a_n comes from a_{n−1}The step.
  3. Walk a few termsThe list appears.

4Arithmetic Progressions

An arithmetic progression has a constant common difference d: 3, 7, 11, 15 has d = 4. The nth term is a + (n−1)d. A list whose gaps drift (3, 7, 12, 18) is not this AP.

Two terms always have a difference; a whole list must keep the same d.

Figure. An arithmetic progression keeps one common difference. 3, 7, 11, 15 has d = 4 every hop, so the nth term is 3 + (n − 1)4.

How it works

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

5Sum of the First n Natural Numbers

The sum of the first n natural numbers 1+2+…+n is n(n+1)/2 — pair first with last (n+1), and there are n/2 such pairs (or the same algebra). For n = 10 the sum is 55, not 10×10.

Adding 1+2+…+10 by a rushed 10×11 is a missing half.

Figure. Pair 1 with 5, 2 with 4, and 3 with itself: five pairs that each make 6 would be too many — it is five numbers, so 5 × 6 / 2 = 15. The two rows are the same list, one reversed.

How it works

  1. Name nHow many naturals.
  2. Write n(n+1)/2The pair-sum.
  3. For n = 10, 10×11/2 = 55A check you can add by hand.

6Geometric Progressions

A geometric progression multiplies by a constant ratio r: 2, 6, 18, 54 has r = 3. The nth term is a r^{n−1}. A list that adds the same d is the other family. r = 1 is a constant list; r = −1 flips sign.

Calling 2, 6, 18 an AP with “growing d” is the family-steal.

Figure. A geometric progression multiplies by one common ratio. 3, 6, 12, 24 has r = 2, so each bar is twice the last.

How it works

  1. Divide consecutive terms (when a ≠ 0)The ratio.
  2. See if the ratio staysGP if yes.
  3. Write a_n = a r^{n−1}The GP rule.

7A definition is a test you can run

Sequence / AP / GP is a test: an ordered list plus a named rule (explicit, recursive, +d, or ×r). 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. The AP test is ‘is every hop the same?’. 5, 8, 11, 14 passes (+3). 5, 8, 12, 17 fails because the hops become +3 then +4.

How it works

  1. Write the ordered list or the ruleThe object.
  2. Name which family and the stepThe test.
  3. Then the word has contentThe definition ran.

8Name the given before the unknown

The given is the first terms or the rule. The unknown is a later term or a sum. Copy a and d (or a and r) before you skip to n = 100.

Using d = 3 because the first term is 3 is a silent given.

Figure. Name a, d and n before you ask for a term. a = 5, d = 3, n = 6 gives a₆ = 5 + 5 × 3 = 20. The unknown is the last box, not the first.

How it works

  1. Copy a, and d or rThe given.
  2. Name a_n or the sum askedThe unknown.
  3. Then plug nGiven first.
3, 7, 11, 15, … is
  1. AP with a = 3, d = 4
  2. GP with r = 4
  3. Not a sequence

Constant gap 4, not a constant ratio.

Notes

  • The official chapter title is “Predicting What Comes Next: Exploring Sequences and 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

  • AP: a_n = a + (n−1)d
  • 1+2+…+n = n(n+1)/2
  • GP: a_n = a r^{n−1}

Recap

Hold these pegs from the official chapter “Predicting What Comes Next: Exploring Sequences and Progressions”. The wording is ExamMaster’s teaching, not a textbook recap.

Introduction to Sequences
A sequence is an ordered list: first term, second term, … The order is the object.
Explicit Rule for a Sequence
An explicit rule names the nth term from n alone: a_n = 3 + (n−1)×4 gives 3, 7, 11, 15, … You do not need the previous term once n is known.
Recursive Rule for a Sequence
A recursive rule names the next from the previous: a1 = 3, a_{n} = a_{n−1} + 4.
Arithmetic Progressions
An arithmetic progression has a constant common difference d: 3, 7, 11, 15 has d = 4.
Sum of the First n Natural Numbers
The sum of the first n natural numbers 1+2+…+n is n(n+1)/2 — pair first with last (n+1), and there are n/2 such pairs (or the same algebra).
Geometric Progressions
A geometric progression multiplies by a constant ratio r: 2, 6, 18, 54 has r = 3.

Practise Predicting What Comes Next: Exploring Sequences and Progressions

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