Board exam focus — Sequences and Series (CBSE & HBSE)

Covers Arithmetic Progression (AP), Geometric Progression (GP), and standard summation formulas. CBSE asks 6–8 marks: nth term, sum to n terms, GP word problem, AM-GM inequality.

Sequence, Series, Arithmetic Progression

Sequence

A sequence is an ordered list of numbers: a₁, a₂, a₃, …, aₙ, … . The n-th term aₙ may be defined by a formula or recurrence.

Series

A series is the sum of terms of a sequence: a₁ + a₂ + … + aₙ + … .

Arithmetic Progression (AP)

A sequence in which the difference between consecutive terms is constant.

If first term is a and common difference is d:

• aₙ = a + (n − 1)d (n-th term)
• Sₙ = (n/2) [2a + (n − 1)d] = (n/2)(a + l), where l = last term

Properties of AP

• Three numbers in AP: a − d, a, a + d.
• Five numbers in AP: a − 2d, a − d, a, a + d, a + 2d.
• Any term = (sum of terms equidistant from it) / 2.
• If each term is multiplied by a constant or a constant is added, the sequence remains an AP.

Arithmetic Mean (AM)

For two numbers a and b: AM = (a + b)/2.

If n means are inserted between a and b, common difference: d = (b − a)/(n + 1).

Sum of First n Natural Numbers

Is an AP with a = 1, d = 1, n terms: Σ k = n(n + 1)/2.

Special Properties

• aₙ − aₙ₋₁ = d for all n ≥ 2.
• Sₙ − Sₙ₋₁ = aₙ for any AP.

CBSE Tip: When asked to find an unknown term or common difference, write two equations from given data and solve simultaneously.

Geometric Progression (GP)

Definition

A sequence in which the ratio of consecutive terms is constant (common ratio r ≠ 0).

If first term = a and common ratio = r:

• aₙ = a · r^(n−1) (n-th term)
• Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1
• Sₙ = na if r = 1

Sum to Infinity

For a GP with |r| < 1: S∞ = a / (1 − r).

This is finite only when |r| < 1; otherwise the series diverges.

Geometric Mean (GM)

For positive numbers a and b: GM = √(ab).

If n geometric means are inserted between a and b, common ratio: r = (b/a)^(1/(n+1)).

AM-GM Inequality

For any two positive numbers a and b: (a + b)/2 ≥ √(ab), with equality iff a = b.

General form (n positive numbers): AM ≥ GM, equality iff all are equal.

Useful Identities

For three numbers a, b, c in GP: b² = ac. For four numbers in GP: bc = ad (and other cross-products).

Properties of GP

• aₙ / aₙ₋₁ = r (constant ratio).
• Each term squared = product of equidistant terms.
• If each term is multiplied by a constant, GP is preserved.

Common Pitfalls

• A GP with negative r alternates in sign.
• Sum formula has r − 1 in denominator — invalid when r = 1; use Sₙ = na instead.

HBSE Tip: When solving for n in Sₙ = k for a GP, you typically get logarithm — keep answer in log form unless calculator is allowed.

Special Sums & Sum of Powers of Naturals

Sum of Powers of First n Natural Numbers

Useful Mental Math

• Σ k = (Σ k³)^(1/2) — yes, the square of the first formula is the third.
• Σ k² > Σ k > n for n ≥ 2.

Splitting Series for Calculation

For sums like Σ (3k² + 2k + 1): = 3 Σ k² + 2 Σ k + Σ 1 = 3 · n(n+1)(2n+1)/6 + 2 · n(n+1)/2 + n.

Simplify, factor out n if possible.

Telescoping Series

A series whose general term can be written as the difference f(k) − f(k − 1), so almost all terms cancel.

Example: Σ 1/[k(k+1)] = Σ [1/k − 1/(k+1)] = 1 − 1/(n+1) = n/(n+1).

Arithmetic-Geometric Progression (AGP)

Terms of the form ARⁿ + (sub-pattern). Sum derived by S − rS shift technique — useful for problems like Σ k · 2^k.

Sum of GP Times AP (Optional)

For series like 1 + 2x + 3x² + 4x³ + … (when |x| < 1): Sum = 1/(1 − x)² (using calculus, not formally proved in Class 11).

Relationship Between AM, GM, HM

For two positive numbers a and b:

• AM = (a + b)/2
• GM = √(ab)
• HM = 2ab/(a + b)

AM ≥ GM ≥ HM, with equality iff a = b.

Also: GM² = AM × HM.

CBSE Tip: Identify Σ k, Σ k², Σ k³ patterns early — split, factor n(n+1), then simplify.

Applications & Word Problems

Common Real-Life Setups

Standard AP Word Problems

• Find total salary after n years
• Days/distances in arithmetic progression
• Stacking objects (triangular numbers, Σ k)

Standard GP Word Problems

• Compound interest: A = P(1 + r/100)^n — GP-like growth
• Half-life / decay: amount halves every τ years
• Bouncing ball: total distance Σ 2hrⁿ + h

Problem-Solving Steps

1. Read carefully — identify AP or GP.
2. List a, d (or r), n.
3. Decide whether n-th term or sum is needed.
4. Apply formula; verify by substituting n = 1 or 2.

Sample Compound Interest as GP

₹10000 at 10% per annum compounded annually for 3 years:

• After year 1: 10000 × 1.1 = 11000
• After year 2: 11000 × 1.1 = 12100
• After year 3: 13310

GP with a = 10000, r = 1.1.

Sample Bouncing Ball

Ball dropped from 16 m bounces to 3/4 of previous height each time. Total distance: 16 + 2(16·3/4 + 16·(3/4)² + …) = 16 + 2·16·(3/4)/(1 − 3/4) = 16 + 96 = 112 m.

HBSE Tip: For "infinite" word problems (bouncing balls, decreasing payments), always check |r| < 1 before using S∞.

Frequently asked questions

Are these Sequences and Series notes free?

Yes — the Sequences and Series notes for Mathematics (Class 11) on Siksha Sarovar are completely free to read, with no account required.

Do these notes follow CBSE and HBSE?

Yes. The Sequences and Series notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.

What does the Sequences and Series chapter cover?

Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Sequences and Series.