8. Predicting What Comes Next: Exploring Sequences and Progressions

9 topics2 h

Chapter 8 of NCERT Mathematics for Class 9: Predicting What Comes Next: Exploring Sequences and Progressions. Four sections — Sequences and the two kinds of rule, Arithmetic progressions, Adding an AP up and Geometric progressions. Nine videos, 2 h in all.

Worked answers to the 35 exercise questions in this chapter →

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Part 1

Sequences and the two kinds of rule

10 minOrder is the point: a sequence carries position as well as value1, 4, 9, 16 has a rule. Shuffle it to 16, 4, 9, 1 and the same four numbers have nothing left to predict.10 min · 12 partsहिंदी9 minAn explicit rule computes any term straight from nWalking to position 1170 costs 1169 additions. A rule that reads the position gets there in two, and would reach the millionth term in two as well.9 min · 12 partsहिंदी10 minA recursive rule builds each term from the ones before it“Add 3 each time” sounds like a rule. It is satisfied by infinitely many different sequences, so something is still missing.10 min · 12 partsहिंदी

Part 2

Arithmetic progressions

10 minCommon difference, and why the nth term is a + (n − 1)dThe (n − 1) is not a correction bolted on to make the numbers work. It counts additions, and there is always one fewer than there are terms.10 min · 12 partsहिंदी9 minAn AP plots as points on a straight lineThe points land on a straight line because a constant difference is exactly what straightness means. What you are looking at is still only dots.9 min · 12 partsहिंदी

Part 3

Adding an AP up

10 minPairing from both ends: a closed form for 1 + 2 + ⋯ + nAdding 1 to 1000 by hand costs 999 additions and one slip ruins the lot. Stop reading the list forwards and it costs one multiplication.10 min · 12 partsहिंदी

Part 4

Geometric progressions

10 minCommon ratio, and why the nth term is ar^(n−1)3, 6, 12, 24 — and the test you have used all chapter fails on it. Stop subtracting consecutive terms and start dividing them.10 min · 12 partsहिंदी9 minFractals: self-similarity generates a GPCut a triangle into four, throw the middle away, repeat forever. The pieces multiply and the area drains away, from one instruction.9 min · 12 partsहिंदी10 minA GP plots as a curve, and what that curve tells youA bent plot rules out an arithmetic progression on sight. It does not rule a geometric one in — the square numbers bend too.10 min · 12 partsहिंदी

What you will be able to do

  • State what makes a list a sequence, and identify the first, second and fifth term of a given sequence
  • Read an expression in n as a rule for the term in position n, and generate the first several terms from it
  • Read a recursive rule as an instruction relating a term to the term before it
  • Test a given sequence for a constant difference and decide whether it is an arithmetic progression
  • Build a stage-and-value table from a growing pattern and read ordered pairs off it
  • Add the first ten counting numbers by pairing from both ends, without adding term by term
  • Test a sequence for a constant ratio and decide whether it is a geometric progression
  • Carry out the Sierpiński triangle construction and draw the next stage from the current one
  • Build a stage-and-value table for a GP and read ordered pairs off it

What this chapter assumes you already know

  • Reading and continuing simple number patterns from Classes 6–8
  • Whole numbers, integers and their arithmetic, including negative differences
  • Square and triangular numbers as figurate counts, met in Class 6
  • Knowing which numbers are counting numbers, and that 94.6 is not one
  • Prime numbers and how they are recognised
  • Substituting into an expression that contains a product, as in 2u + 3
  • Reading an inequality such as n ≥ 2 as a condition on where a rule applies
  • Arithmetic with negative numbers, for the exercise that starts at −5

It builds directly on What "rational" means, and why the denominator cannot be zero, Two axes, an origin, and why the order of the pair matters, Turning a situation into an expression: terms, variables, coefficients, What makes a polynomial linear, and the equation you get by fixing its value, A constant difference is the signature of a linear pattern and What a and b do to the line: slope, y-intercept, and parallel families.

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "A sequence is just a set of numbers."
  • "The three dots mean a few more numbers follow."
  • "t₄ means t multiplied by 4."
  • "Different letters mean different kinds of sequence."
  • "Sequences go up."
  • "The position can be 0, because the chapter says a position is a non-negative integer."
  • "A triangular number is a triangle."
  • "You still have to know the earlier terms."