Sequences and Series
Class 11 · Mathematics
4 ideas in Class 11 Mathematics, explained in plain words, with games and quizzes to practise.
Sequences and Series Basics
Part of: Sequences and Progressions
Sequences order terms finitely or infinitely; rules like a-n formulas generate them; adding terms builds series; Fibonacci adds previous pairs.
Why it matters. Term rules predict values. Series total sequences.
Formulas
- Series sum S(n) = a(1) + ... + a(n)
- Fibonacci: next term = sum of previous two
Key words
- Sequence
- : Ordered list of terms, finite or infinite.
- Series
- : Total built by adding sequence terms.
- General term
- : Rule like a(n) = 3n - 4 generating each term.
- Finite sequence
- : List ending after a fixed count of terms.
- Infinite sequence
- : List running on without end.
- Fibonacci sequence
- : List where each term adds the previous two.
Try it in two minutes
- Substitute n for terms.
- Spot finite versus infinite.
- Add for the series.
Try at home. Write first five triangle numbers.
Arithmetic Progression
Part of: Sequences and Progressions
Constant difference defines A.P.; nth term adds n-minus-one differences; sums average first-last times count; last-term and zero-sum tricks shortcut.
Why it matters. Steady steps model growth. Sums total plans.
Formulas
- a(n) = a + (n - 1)d
- S(n) = n/2 x (2a + (n - 1)d)
- S(n) = n/2 x (first + last)
Key words
- Arithmetic progression
- : List with a constant gap between neighbours.
- Common difference
- : Constant gap added at every step.
- nth term
- : Term built as a plus n-minus-one gaps.
- Sum shortcut
- : Average of first and last, times the count.
- Finding d
- : Gap found from any two known terms.
Try it in two minutes
- Find the difference.
- Build nth term.
- Average ends for sum.
Try at home. Sum fares 20, 25, 30 for 10 days.
Geometric Progression
Part of: Geometric Means and Sums
Constant ratio defines G.P.; nth term multiplies a by r-to-n-minus-one; finite sums divide a-times-one-minus-r-to-n; infinite sums need modulus-r under one.
Why it matters. Ratios model doubling. Infinite sums converge conditionally.
Formulas
- a(n) = a x r^(n - 1)
- S(n) = a x (r^n - 1) / (r - 1)
- Infinite sum = a / (1 - r) when |r| < 1
Key words
- Geometric progression
- : List with a constant multiplier between neighbours.
- Common ratio
- : Constant multiplier applied at every step.
- nth term
- : Term built as a times r to n-minus-one.
- Infinite sum
- : Sum that settles only when the ratio modulus is below one.
- Alternating G.P.
- : Ratio minus one flipping signs term by term.
Try it in two minutes
- Find the ratio.
- Raise for nth term.
- Check ratio size for infinity.
Try at home. Halve a strip five times.
Means and Special Sums
Part of: Geometric Means and Sums
A.M. averages, G.M. root-multiplies, A.M. never trails G.M.; sums of n, n-squared, n-cubed follow closed formulas; telescoping splits cancel middle.
Why it matters. Means bound growth. Power sums total series fast.
Formulas
- A.M. = (a + b) / 2 and G.M. = sqrt(ab)
- A.M. >= G.M. for positive numbers
- Sum 1..n = n(n + 1)/2
- Sum squares = n(n + 1)(2n + 1)/6
- Sum cubes = [n(n + 1)/2]^2
Key words
- Arithmetic mean
- : Plain average of two numbers.
- Geometric mean
- : Root of the product of two numbers.
- AM-GM rule
- : A.M. never trails G.M. for positive numbers.
- Telescoping sum
- : Split whose middle terms cancel in pairs.
- Special sums
- : Closed totals for powers one, two and three.
Try it in two minutes
- Average versus multiply.
- Apply A.M.-G.M. order.
- Plug power-sum formulas.
Try at home. Compare mean heights two ways.
What your child will practise
- Arithmetic Progression: Ocean Puzzle (Clue Puzzle)
- Arithmetic Progression: Ocean Quick Fire (Quick Fire)
- Geometric Progression: Jungle Puzzle (Clue Puzzle)
- Means and Special Sums: Desert Quest (Flip and Match)
- Sequences and Series Basics: Desert Quick Fire (Quick Fire)
- Quiz: Arithmetic Progression (8 questions)
- Quiz: Geometric Progression (8 questions)
- Quiz: Means and Special Sums (8 questions)
- Quiz: Sequences and Series Basics (8 questions)