8. Sequences and Series

8 topics2 h

Chapter 8 of NCERT Mathematics for Class 11: Sequences and Series. Three sections — Ordered lists, and their running totals, Multiplying by the same factor each step and Adding infinitely many terms. Eight videos, 2 h in all.

Worked answers to the 64 exercise questions in this chapter: Exercise 8.1 · Exercise 8.2 · Miscellaneous Exercise

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Part 1

Ordered lists, and their running totals

13 minA rule that turns a position number into a termCounting ancestors back three centuries, thirty years to a generation, gives ten answers and stops. Dividing ten by three, stage by stage, never stops at all.13 min · 12 parts11 minWhat changes once the terms are added instead of listed1 + 3 + 5 + 7 has four terms in it. 16 has none. They are not the same object, and almost every difficulty in this topic comes from treating them as though they were. Six different series in the explanation all come to 16 - seventeen terms between them and one number at the end of all six - and no amount of staring at the 16 tells you which one it came from.11 min · 10 parts

Part 2

Multiplying by the same factor each step

12 minA constant ratio between neighbours is the entire definitionA geometric progression is not a shape you learn to recognise. It is a verdict you earn by dividing - and the difference is measurable. Eleven sequences are put to both judges here: an eye that looks for terms climbing by a whole multiple, and a test that divides at every step. They disagree about eight of the eleven. The eye walks past seven real progressions and waves one impostor through.12 min · 12 parts14 minReaching any term without walking through the earlier onesStart at two, multiply by three, over and over: two, six, eighteen, fifty-four. Reaching position sixteen costs fifteen multiplications — the opening term starts unmultiplied.14 min · 12 parts15 minSubtracting a scaled copy of the total to collapse it to two termsWrite a geometric progression's sum, then that sum times its own ratio, underneath. Slide one line along and the middle terms match their neighbours — subtract, and they cancel.15 min · 13 parts13 minThe number that sits between two others multiplicativelyAn empty box sits between two and eight, and nothing belongs there until a job is named: the same multiplier in as out. The box holds four, or minus four — never the average, five.13 min · 15 parts10 minWhy one of the two means can never overtake the otherFour and sixteen: the ordinary average is ten, the geometric middle is eight, and the average always wins — the gap is half the square of a difference of square roots.10 min · 13 parts

Part 3

Adding infinitely many terms

14 minWhen the ratio is small enough, an endless sum still settles on a numberNo finite total of 1, two-thirds, four-ninths and onward ever reaches 3, and every one falls short forever. The sum to infinity is exactly 3 all the same.14 min · 14 parts

What you will be able to do

  • Distinguish the position number from the term standing at that position, and write both using the chapter's subscript notation
  • State the difference between a series and the number its addition yields, and use the two words correctly
  • Test a given sequence for constant ratio by dividing consecutive terms, and state the verdict with the working shown
  • Derive the rule for the term at position n of a G.P. by counting multiplications, rather than quoting it
  • Derive a G.P.'s total over its opening n terms by writing the scaled copy and then subtracting, without quoting the result
  • State the condition that defines the geometric mean of two positive numbers, and derive the square-root formula from it
  • State both means of a pair of positive numbers and compute each
  • Explain what it means for an endless addition to have a sum, in terms of the finite totals rather than in terms of adding forever

What this chapter assumes you already know

  • The natural numbers as an ordered counting set, and the idea of a subset of them
  • Function notation: an input, an output, and the requirement that one input gives one output
  • Substituting a number for a letter in an algebraic expression, including expressions with a fraction bar and with a power
  • Signed arithmetic — multiplying three signed factors, and reading a power of −1 as a sign switch
  • Reading subscript notation, which the chapter uses from its second page onward
  • Reading a rule that refers back to earlier terms, and unwinding it in order
  • Addition of signed numbers and of fractions with different denominators
  • Comfort with a letter standing for a count that has not been fixed yet

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.

  • "If you cannot write a formula, it is not a sequence."
  • "The terms have to increase."
  • "The terms have to be different from each other."
  • "aₙ and n are interchangeable."
  • "Infinite means the values get arbitrarily large."
  • "A backward-looking rule can be evaluated at any position directly."
  • "The first term is a₀."
  • "Series and sum are two words for one thing."