Chapter 9 — Sequences And Series

What is a sequence?
A sequence is an ordered list of numbers written as a1, a2, a3, ... , an, ... A sequence can be finite or infinite. Each number in the list is called a term of the sequence. We usually write a sequence as {an} or (an) where n is the index (natural number) and an is the nth term (general term).

General term and recurrence
The general term an (also called nth term) gives the value of the term in terms of n. Many sequences are given by a recurrence relation: an is defined using previous terms (for example an = an-1 + d).

Important types

• Arithmetic Progression (AP): consecutive terms differ by a constant d (common difference). Example: 5, 8, 11, 14, ...
• Geometric Progression (GP): consecutive terms have a constant ratio r (common ratio). Example: 3, 6, 12, 24, ...
• Harmonic Progression (HP): reciprocals of the terms form an AP. Example: 1, 1/2, 1/3, 1/4, ... is not HP; but 1, 1/3, 1/5, 1/7, ... is an HP because reciprocals 1, 3, 5, 7 are in AP.

Monotonicity and boundedness
A sequence is increasing if an < an+1 for all n, decreasing if an > an+1. For AP: if d > 0 increasing, d < 0 decreasing. For GP with positive a: if r > 1 increasing, 0 < r < 1 decreasing. Boundedness: AP is unbounded unless d = 0; GP is bounded if |r| < 1.

Means and relations
Arithmetic mean between two numbers a and b is (a+b)/2. Geometric mean is sqrt(ab). For three numbers in AP: 2b = a + c. For three numbers in GP: b^2 = ac.

Why sequences matter (intuition and applications)
Sequences describe many stepwise processes: repeated payments, population growth models, savings with regular deposits, decaying quantities, and arrangement patterns. They are the building blocks of series (sums of sequences) and are foundational for limits and calculus.

Short derivation (sum of first n terms of AP)
If AP is a, a+d, ..., a+(n-1)d, sum S_n = n/2 [first + last] = n/2 [2a + (n-1)d]. (Pairing first with last gives constant sums.)

Short derivation (sum of first n terms of GP)
For GP a, ar, ar^2, ..., ar^{n-1}, multiply S_n by r and subtract to get S_n(1-r) = a(1-r^n). Hence S_n = a(1-r^n)/(1-r) for r ≠ 1. For |r| < 1, as n → ∞ the infinite sum S = a/(1-r).

What is a series? A series is the sum of terms of a sequence. If {a_n} is a sequence, then the expression a_1 + a_2 + a_3 + … + a_n is called the finite series (or sum to n terms) and a_1 + a_2 + a_3 + … continuing indefinitely is called an infinite series.

Notation (Sigma): A finite series is often written using sigma notation as Σ_{k=1}^{n} a_k. This compactly represents the sum of terms a_1, a_2, ..., a_n.

Types of series studied in Class 11

• Arithmetic series (AP series): when the underlying sequence is an arithmetic progression with first term a and common difference d, the nth term is a_n = a + (n-1)d and the sum of n terms is S_n = n/2 [2a + (n-1)d] = n(a + a_n)/2.
• Geometric series (GP series): when the sequence is geometric with first term a and common ratio r, the nth term is a_n = a r^{n-1} and the sum of n terms is S_n = a(1 - r^n)/(1 - r) for r ≠ 1. For an infinite GP with |r| < 1, the sum to infinity is S_∞ = a/(1 - r).
• Telescoping series: series in which many terms cancel out when written in expanded form; used to find finite sums easily.

Convergence of infinite series (basic idea): An infinite series a_1 + a_2 + … converges if the sequence of partial sums S_n = Σ_{k=1}^{n} a_k approaches a finite limit as n → ∞. For Class 11 you primarily use this test for geometric series: it converges only when |r| < 1.

Useful standard sums: sums of first n natural numbers, squares and cubes are frequently used in problems and proofs.

Strategy for solving problems: identify the type of progression (AP, GP, or telescoping), write the nth term, use the appropriate sum formula or cancellation, and check convergence conditions for infinite sums.

What is Sigma (Summation) Notation?

Sigma notation uses the Greek letter Σ to represent a sum of terms of a sequence compactly. The general form is Σ_i=mⁿ a_i, which means a_m + a_m+1 + ... + a_n. The index i is a dummy variable that runs from the lower limit m to the upper limit n. Summation usually denotes a finite sum in Class 11 topics.

Basic points to read the notation

• Lower limit (m): starting index.
• Upper limit (n): ending index.
• a_i: general term (may depend on i).
• Dummy index: naming it i, k, j doesn't change the value as long as it's used consistently.

Key properties of finite sums

• Linearity (sum of sums): Σ_i=mⁿ (x_i + y_i) = Σ_i=mⁿ x_i + Σ_i=mⁿ y_i.
• Constant factor: Σ_i=mⁿ c·x_i = c · Σ_i=mⁿ x_i for constant c.
• Index shift / reindexing: Σ_i=mⁿ a_i = Σ_j=m+k^n+k a_j-k. You may shift the index to simplify.
• Splitting the range: Σ_i=mⁿ a_i = Σ_i=m^p a_i + Σ_i=p+1ⁿ a_i for any m ≤ p < n.
• Sum of constant: Σ_i=mⁿ c = c·(n - m + 1).
• Telescoping sums: If terms cancel successively, e.g., Σ_k=1ⁿ (b_k - b_k+1) = b₁ - b_n+1.

Partial sums and interpretation

Given a sequence {a_n}, the nth partial sum S_n = Σ_k=1ⁿ a_k. Studying S_n helps understand accumulation (like total distance, total cost) and, for infinite series, convergence (Class 11 introduces finite sums and prepares for infinite series later).

Useful specific summation formulas

Some frequently used closed forms (for n a positive integer):

• Σ_i=1ⁿ i = n(n + 1)/2
• Σ_i=1ⁿ i² = n(n + 1)(2n + 1)/6
• Σ_i=1ⁿ i³ = [n(n + 1)/2]² (i.e., (sum of first n integers)²)
• For geometric sequence: Σ_i=0ⁿ arⁱ = a(1 - rⁿ⁺¹)/(1 - r) for r ≠ 1
• Arithmetic progression sum: If a, a + d, ..., a + (n-1)d are terms, Σ_k=0^n-1 [a + kd] = n/2 · [2a + (n - 1)d]

Why the notation is useful

Sigma notation compresses long repeated addition into a formula, makes algebraic manipulation easier (use properties to factor, split, or simplify sums), and connects sequences to geometry (areas under step functions) and applications such as totals, averages, and financial accumulations.

Definition: An arithmetic progression (AP) is a sequence of numbers in which the difference between any two successive terms is constant. This constant is called the common difference and is usually denoted by d. If the first term is a, the AP is: a, a + d, a + 2d, a + 3d, ...

General (nth) term: The nth term (denoted a_n) of an AP is given by

a_n = a + (n − 1)d

This formula is obtained by adding the common difference (n − 1) times to the first term.

Sum of first n terms: The sum S_n of the first n terms of an AP can be written as

S_n = n/2 [2a + (n − 1)d] = n/2 (a + l)

where l is the nth (or last) term, l = a + (n − 1)d. The second form uses the average of the first and last term (arithmetic mean) multiplied by the number of terms.

Properties and useful results:

• The common difference d = a₂ − a₁ = a₃ − a₂ = ...
• If three numbers p, q, r are consecutive terms of an AP, then q = (p + r)/2 (q is arithmetic mean of p and r).
• To check whether a sequence is an AP, verify that successive differences are equal (first differences constant); equivalently, second differences are zero.
• If the last term l is known, the number of terms is n = ((l − a)/d) + 1 (when d ≠ 0).
• If n is odd, the middle term equals the average (mean) of the AP and equals S_n/n.

Derivation sketch of S_n: Write S_n = a + (a + d) + ... + (a + (n − 1)d). Write the sum again in reverse order and add both expressions termwise; each pair gives (a + l). There are n such pairs, giving 2S_n = n(a + l) → S_n = n/2(a + l). Replacing l by a + (n − 1)d yields the alternative form.

Definition: A sequence is called a Geometric Progression (GP) if the ratio of any term to its preceding term is constant. This constant is called the common ratio (r). If a is the first term, the GP is:
a, a·r, a·r², a·r³, …

General term (nth term):
The nth term t_n of a GP with first term a and common ratio r is
t_n = a·r^n-1.

Sum of first n terms (S_n):
For r != 1, let S_n = a + a·r + a·r² + … + a·r^n-1. Multiply by r and subtract:

1. r·S_n = a·r + a·r² + … + a·rⁿ
2. Subtract: (1 − r)S_n = a − a·rⁿ

So, S_n = a(1 − rⁿ)/(1 − r) for r != 1. If r = 1, then S_n = n·a.

Infinite GP:
If |r| < 1, rⁿ → 0 as n → ∞, so the infinite sum converges to
S = a/(1 − r). If |r| ≥ 1, the infinite series diverges (no finite sum).

Properties and remarks:

• Consecutive-term ratio: r = t_n+1/t_n (provided t_n ≠ 0).
• If r > 0, terms keep the same sign; if r < 0, terms alternate in sign.
• Geometric mean of two positive numbers b and c is √(b·c). In a GP, each term (except first and last) is the geometric mean of its neighbours: r = t_n+1/t_n = t_n/t_n−1.
• Product of first n terms: P_n = aⁿ·r^n(n−1)/2.

When to use GP: GP models processes where every step multiplies the previous value by a constant factor — e.g., compound growth/decay, repeated proportional change, binary fission (doubling).

Definition: A sequence {a_n} is called a Harmonic Progression (HP) if the reciprocals {1/a_n} form an Arithmetic Progression (AP). In other words, a_n is in HP ⇔ 1/a_n is in AP (for all terms a_n ≠ 0).

Understanding: If 1/a_1, 1/a_2, 1/a_3, ... is an AP with first term A and common difference d, then 1/a_n = A + (n-1)d, so a_n = 1 / (A + (n-1)d).

Key properties:

• Three numbers a, b, c are in HP iff their reciprocals are in AP, which gives the condition 2/b = 1/a + 1/c (equivalently b = 2ac/(a + c)).
• The harmonic mean (HM) of two numbers a and b is HM = 2ab/(a + b). For n positive numbers x_1, x_2, ..., x_n, the harmonic mean is HM = n / (Σ(1/x_i)).
• Sum of first n terms of an HP cannot be written in a simple closed form in general; it is S_n = Σ_{k=1}^n 1/(A + (k-1)d) which is the sum of reciprocals of an AP.
• Relation between means: For positive numbers, AM ≥ GM ≥ HM (Arithmetic Mean ≥ Geometric Mean ≥ Harmonic Mean).
• Special example: the sequence 1, 1/2, 1/3, ... is an HP (the harmonic sequence). Its series Σ 1/n (the harmonic series) diverges, though the terms themselves tend to 0.

How to generate an HP from an AP: Given an AP with terms A, A + d, A + 2d, ..., take reciprocals to get an HP: 1/A, 1/(A + d), 1/(A + 2d), ....

Definitions (for n positive numbers a1, a2, ..., an):

• Arithmetic Mean (AM) = (a1 + a2 + ... + an)/n.
• Geometric Mean (GM) = (a1·a2·...·an)^(1/n), defined for ai > 0.
• Harmonic Mean (HM) = n / (1/a1 + 1/a2 + ... + 1/an), defined for ai > 0.

Main relation: For any positive real numbers a1, a2, ..., an,

AM ≥ GM ≥ HM,

Proof sketch:

• AM ≥ GM for two numbers a and b: (a − b)^2 ≥ 0 ⇒ a^2 + b^2 ≥ 2ab ⇒ (a + b)^2 ≥ 4ab ⇒ (a + b)/2 ≥ √(ab). So AM ≥ GM for n = 2. General n can be proved by induction or by repeated application of the two-variable case (standard AM–GM proof).
• GM ≥ HM: Apply AM ≥ GM to the reciprocals 1/a1, ..., 1/an. Then (1/n)∑(1/ai) ≥ (∏(1/ai))^(1/n) = 1/GM. Multiply both sides by n and take reciprocals (inequality reverses when taking reciprocals of positive numbers) to get HM = n/∑(1/ai) ≤ GM.
• Combining gives AM ≥ GM ≥ HM, equality only when all ai are equal.

Intuition: AM is influenced by large values (linear averaging), GM reflects multiplicative/relative central tendency (useful for growth rates), and HM gives more weight to small values — appropriate when averaging rates like speeds or resistances in parallel.

What are Special Summation Formulas? In Sequences and Series, special summation formulas give closed forms for sums of powers of the first n natural numbers. They let us evaluate sums like Σr, Σr^2, Σr^3 quickly without adding every term.

Common formulas (for n a positive integer):

• Σ_r=1ⁿ r = n(n+1)/2
• Σ_r=1ⁿ r^2 = n(n+1)(2n+1)/6
• Σ_r=1ⁿ r^3 = [n(n+1)/2]^2

Why they work (brief proofs):

• Sum of first n natural numbers: pair terms from ends: (1+n), (2+n-1), … gives n/2 pairs each of size (n+1), so sum = n(n+1)/2.
• Sum of squares: can be proved by mathematical induction or by known polynomial fitting (the sum is a cubic polynomial in n; determine coefficients using base values).
• Sum of cubes identity: (1 + 2 + … + n)^2 = 1^3 + 2^3 + … + n^3. This is proved by induction or algebraic manipulation and gives Σr^3 = [n(n+1)/2]^2.

Useful derived formulas (obtained by linearity of summation):

• Σ r(r+1) = Σ r^2 + Σ r = n(n+1)(n+2)/3
• Σ r(r+1)(r+2) = n(n+1)(n+2)(n+3)/4
• Sum of an AP: a + (a+d) + … + [a+(n-1)d] = n/2[2a+(n-1)d]

How to use them: Replace sums of powers by the formulas to simplify algebraic expressions, evaluate finite series, or compute combinatorial counts (handshakes, seating arrangements, etc.).