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NCERT · Ganita Manjari · Volume 1 · Chapter 8

Predicting What Comes Next: Exploring Sequences and Progressions

53 videos in this chapter

Sequences and their notation

5 videos

Chapter 8A Sequence Is an Ordered List of Terms3:48A Sequence Is an Ordered List of TermsYou'll learn: Explain position, term value, ellipsis and finite versus infinite sequences.Chapter 8Build Triangular Numbers by Adding the Next Row3:35Build Triangular Numbers by Adding the Next RowYou'll learn: Connect dot arrays with cumulative sums of natural numbers.Chapter 8Odd Borders Build the Square Numbers3:38Odd Borders Build the Square NumbersYou'll learn: Connect successive odd additions with enlarging square arrays.Chapter 8A New Sequence Can Come from Running Totals3:50A New Sequence Can Come from Running TotalsYou'll learn: Generate partial sums and compare them with the original terms.Chapter 8Read Subscripts and Include Negative or Fractional Terms3:45Read Subscripts and Include Negative or Fractional TermsYou'll learn: Use tₙ, sₙ, uₙ and identify increasing, decreasing or signed sequences.

Explicit and recursive rules

8 videos

Chapter 8An Explicit Rule Finds a Term Directly from Its Position3:36An Explicit Rule Finds a Term Directly from Its PositionYou'll learn: Substitute a positive integer index into a term formula.Chapter 8Work Backwards to Find a Term’s Position3:41Work Backwards to Find a Term’s PositionYou'll learn: Solve a term equation and test whether the index is a positive integer.Chapter 8Generate Sequences from Linear and Quadratic Rules3:50

Arithmetic progressions

5 videos

Chapter 8Recognise an Arithmetic Progression by Its Constant Difference3:50Recognise an Arithmetic Progression by Its Constant DifferenceYou'll learn: Compare successive differences and identify the special class of sequences.Chapter 8Why an AP’s nth Term Uses n−1 Steps3:35Why an AP’s nth Term Uses n−1 StepsYou'll learn: Count the jumps from the first term to derive tₙ=a+(n−1)d.Chapter 8APs Can Decrease or Have Fractional Terms3:43

AP applications and recovery problems

3 videos

Chapter 8Find a Distant AP Term or Test Membership3:28Find a Distant AP Term or Test MembershipYou'll learn: Use the general term forwards and backwards.Chapter 8Recover an AP from Two Term Conditions3:43Recover an AP from Two Term ConditionsYou'll learn: Use differences between term equations to find a and d.Chapter 8Taxi Fares and Salaries Can Form APs3:43Taxi Fares and Salaries Can Form APs

Sums of natural numbers and triangular numbers

4 videos

Chapter 8Pair a Sum with Its Reverse3:37Pair a Sum with Its ReverseYou'll learn: Add1–10 in opposite directions so every column has the same total.Chapter 8Derive the Sum of the First n Natural Numbers3:36Derive the Sum of the First n Natural NumbersYou'll learn: Generalise the reversed-sum proof and connect it to a rectangular dot array.Chapter 8Find a Sum That Starts Somewhere Other Than One3:50

Natural-number sums and AP applications

1 video

Chapter 8Count and Add the Two-Digit Multiples of Three3:35Count and Add the Two-Digit Multiples of ThreeYou'll learn: Identify the first and last terms, count the steps and add the required multiples.

Geometric progressions

5 videos

Chapter 8A GP Multiplies Where an AP Adds3:54A GP Multiplies Where an AP AddsYou'll learn: Compare equal differences with equal ratios using the two growth pictures.Chapter 8Why a GP’s nth Term Has Exponent n−13:35Why a GP’s nth Term Has Exponent n−1You'll learn: Count repeated multiplications to derive tₙ=arⁿ⁻¹.Chapter 8Find the Common Ratio in Growing and Shrinking GPs3:42Find the Common Ratio in Growing and Shrinking GPs

GP term applications

2 videos

Chapter 8Use a Known GP Term to Find a Later Term3:31Use a Known GP Term to Find a Later TermYou'll learn: Move a fixed number of positions by multiplying by the ratio that many times.Chapter 8Which GP Term Equals a Given Number?3:34Which GP Term Equals a Given Number?You'll learn: Match the target with a power of the common ratio.

Fractals and geometric patterns

6 videos

Chapter 8Build the Sierpiński Triangle One Removal at a Time4:00Build the Sierpiński Triangle One Removal at a TimeYou'll learn: Join side midpoints and remove each central small triangle repeatedly.Chapter 8Count the Small Triangles with a Power of Three3:41Count the Small Triangles with a Power of ThreeYou'll learn: Explain why each retained triangle produces three at the next stage.Chapter 8More Triangles Can Occupy Less Total Area3:41

Bouncing-ball applications

2 videos

Chapter 8A Bouncing Ball’s Heights Form a Shrinking GP3:50A Bouncing Ball’s Heights Form a Shrinking GPYou'll learn: Generate bounce maxima and find the first bounce below a specified threshold.Chapter 8Count Both Upward and Downward Travel Between Bounces3:31Count Both Upward and Downward Travel Between BouncesYou'll learn: Distinguish rebound height from total vertical distance travelled.

Chapter challenge problems

12 videos

Chapter 8A Shifted Recursive Sequence Can Reveal a GP3:28A Shifted Recursive Sequence Can Reveal a GPYou'll learn: Transform the recurrence into a geometric rule and identify a target term.Chapter 8Count Multiples Within a Fixed Number Range3:46Count Multiples Within a Fixed Number RangeYou'll learn: Use the smallest and largest allowable multiples and the AP term formula.Chapter 8Recover an AP from Sums of Two Terms3:41Recover an AP from Sums of Two Terms

Generate Sequences from Linear and Quadratic Rules

You'll learn: Evaluate several explicit rules and compare the resulting sequences.

Chapter 8A Recursive Rule Needs a Starting Value3:40A Recursive Rule Needs a Starting ValueYou'll learn: Generate terms from the preceding term and compare with an explicit description.

Chapter 8Use a Recursive Machine That Doubles and Adds Three3:43Use a Recursive Machine That Doubles and Adds ThreeYou'll learn: Calculate successive terms and check a proposed value by continuing the rule.

Chapter 8A Recursive Rule Can Grow Much Faster Than Addition3:41A Recursive Rule Can Grow Much Faster Than AdditionYou'll learn: Evaluate a rule that multiplies the previous term by one less than itself.

Chapter 8The Next Virahanka–Fibonacci Term Uses the Previous Two3:46The Next Virahanka–Fibonacci Term Uses the Previous TwoYou'll learn: Generate the sequence from two initial values and explain why both are needed.

Chapter 8A Rule Can Use the Previous Three Terms3:46A Rule Can Use the Previous Three TermsYou'll learn: Apply a three-term recurrence and contrast its initial conditions with the two-term rule.

APs Can Decrease or Have Fractional Terms

You'll learn: Identify a and d in positive, negative, fractional and decimal examples.

Chapter 8Plot an AP as Discrete Points on a Straight Line3:56Plot an AP as Discrete Points on a Straight LineYou'll learn: Turn a term table into ordered pairs(index, value).

Chapter 8Write the Same AP Explicitly and Recursively3:48Write the Same AP Explicitly and RecursivelyYou'll learn: Translate between tₙ=a+(n−1)d and tₙ=tₙ₋₁+d.

You'll learn: Identify the first observed term and the fixed increment in a context.

Find a Sum That Starts Somewhere Other Than One

You'll learn: Subtract two initial sums to retain exactly the intended interval.

Chapter 8The nth Triangular Number Is a Sum3:31The nth Triangular Number Is a SumYou'll learn: Connect the original triangle arrays to the closed term formula.

You'll learn: Divide consecutive non-zero terms and derive term formulas.

Chapter 8Negative Ratios Can Make the Signs Alternate3:53Negative Ratios Can Make the Signs AlternateYou'll learn: Generate terms and verify a signed ratio.

Chapter 8Describe a GP with a Recursive Rule3:51Describe a GP with a Recursive RuleYou'll learn: Translate the explicit expression into a repeated-multiplier rule.

More Triangles Can Occupy Less Total Area

You'll learn: Use the retained fraction to model the shrinking shaded area.

Chapter 8Plot AP and GP Behaviour Side by Side3:55Plot AP and GP Behaviour Side by SideYou'll learn: Compare constant additive change with multiplicative growth and decay.

Chapter 8Build and Count a Sierpiński Square Carpet3:41Build and Count a Sierpiński Square CarpetYou'll learn: Remove the central ninth in every retained square and count the remaining pieces.

Chapter 8Find the Shrinking Area of the Square Carpet3:42Find the Shrinking Area of the Square CarpetYou'll learn: Apply the retained fraction8/9 at every stage.

You'll learn: Convert term-sum information into two linear conditions on a and d.

Chapter 8When Does a Natural-Number Sum First Exceed One Thousand?3:51When Does a Natural-Number Sum First Exceed One Thousand?You'll learn: Use the triangular-sum formula and test neighbouring integer positions.

Chapter 8Write One Hundred as Consecutive Natural Numbers3:37Write One Hundred as Consecutive Natural NumbersYou'll learn: Use a starting value and number ofterms to find every valid representation.

Chapter 8A Doubling Bacterial Culture Uses Stage Zero3:43A Doubling Bacterial Culture Uses Stage ZeroYou'll learn: Model the initial count and subsequent hourly doublings.

Chapter 8Find a GP from an Initial Sum and Later-Term Relation3:49Find a GP from an Initial Sum and Later-Term RelationYou'll learn: Express the two conditions using a and r and find possible progressions.

Chapter 8Find Three GP Terms from Their Sum and Product3:51Find Three GP Terms from Their Sum and ProductYou'll learn: Use the middle term and equal ratios to constrain the first and third terms.

Chapter 8Equally Spaced GP Positions Also Form a GP3:33Equally Spaced GP Positions Also Form a GPYou'll learn: Compare the ratios between terms separated by the same number of steps.

Chapter 8Recover Three GP Terms from a Sum and Sum ofSquares3:39Recover Three GP Terms from a Sum and Sum ofSquaresYou'll learn: Combine algebraic identities with the GP middle-term relation.

Chapter 8A Sum-Based Recurrence Can Collapse to Doubling3:38A Sum-Based Recurrence Can Collapse to DoublingYou'll learn: Subtract consecutive recurrence statements to simplify the rule.

Chapter 8Find a Familiar Sequence Hidden in Another Recurrence3:44Find a Familiar Sequence Hidden in Another RecurrenceYou'll learn: Compare consecutive cumulative-sum rules and recognise a two-term recurrence.