Sequences and Series | IISER Smart Prep

1. Core Concept

A sequence is an ordered list of numbers following a mathematical rule. Arithmetic Progressions (AP) grow by repeatedly adding a constant difference, while Geometric Progressions (GP) grow by repeatedly multiplying by a constant ratio.

2. Key Formulas

Arithmetic Progression (AP)

Given first term = a, common difference = d.

n^th term: t_n = a + (n - 1)d
Sum of n terms: S_n = (n/2)[2a + (n - 1)d] = (n/2)(a + l)

(where l is the last term)

Geometric Progression (GP)

Given first term = a, common ratio = r.

n^th term: t_n = ar^n-1
Sum of n terms: S_n = a(1 - rⁿ) / (1 - r) (for r ≠ 1)
Sum of Infinite GP: S_∞ = a / (1 - r) (Valid ONLY if |r| < 1)

Means

Arithmetic Mean (AM): (a + b) / 2
(For n terms: (x₁ + x₂ + … + x_n) / n)
Geometric Mean (GM): √(ab)
(For n terms: (x₁ · x₂ … x_n)^1/n)

Insertion of Means

Insert n AMs between a and b:
Common difference d = (b - a)/(n + 1)
Insert n GMs between a and b:
Common ratio r = (b/a)^{1/(n + 1)}

Sum of Special Series

Sum of first n natural numbers: ∑k = n(n + 1) / 2
Sum of squares: ∑k² = n(n + 1)(2n + 1) / 6
Sum of cubes: ∑k³ =[n(n + 1) / 2]² = (∑k)²

3. Important Results

AM ≥ GM

AM-GM Inequality: For any set of positive real numbers, the Arithmetic Mean is always greater than or equal to the Geometric Mean. (Equality holds ONLY when all numbers in the set are equal).

Basic Conditions

If a, b, c are in AP &implies; 2b = a + c
If a, b, c are in GP &implies; b² = ac

Selection of Terms (Shortcut)

3 terms in AP: a - d, a, a + d (Sum cancels d)
4 terms in AP: a - 3d, a - d, a + d, a + 3d (Common difference is 2d)
3 terms in GP: a/r, a, ar (Product cancels r)

4. Conceptual Insights

General Term from Sum: If you are given the formula for the sum of n terms (S_n), you can instantly find the n^th term using: t_n = S_n - S_n-1.
Linear vs. Exponential: An AP represents linear growth (like simple interest), while a GP represents exponential growth/decay (like compound interest or radioactive decay).
Logarithmic Link: If a, b, c are positive numbers in GP, then log(a), log(b), log(c) are in AP!

Why |r| < 1?

If |r| < 1, terms keep getting smaller → series converges to a finite value.

If |r| ≥ 1, terms do not shrink → sum becomes infinite (diverges).

5. Common Mistakes

Infinite GP Trap: Applying S_∞ = a / (1 - r) when r ≥ 1 or r ≤ -1. The series diverges in these cases; there is no finite sum.
AM-GM on Negative Numbers: AM ≥ GM is strictly for positive numbers. If you apply it to negative numbers, the square root for GM might become imaginary, or the inequality will fail entirely.
Miscounting 'n': In series like 2² + 3² + … + 10², students often blindly use n = 10 in the formula. The standard formula strictly starts from 1². You must calculate (∑_k=1¹⁰ k²) - 1².

6. Example Applications

Optimization without Calculus

Q: "Find the minimum value of (x + 1/x) for x > 0."

Insight: Use AM ≥ GM.

(x + 1/x) / 2 ≥ √(x · 1/x) &implies; (x + 1/x) / 2 ≥ 1 &implies; x + 1/x ≥ 2

Minimum value is 2 (occurs when x = 1/x &implies; x = 1).

Method of Differences (Telescoping)

Q: Sum 1/(1·2) + 1/(2·3) + … + 1/(n(n+1)).

Insight: Rewrite the general term as partial fractions: 1/(k(k+1)) = 1/k - 1/(k+1).

When you sum them, all middle terms cancel out in a chain reaction, leaving just the first and last pieces:
1 - 1/(n+1) = n/(n+1).

7. IAT Exam Focus Points

AM-GM Inequality (Top Tier): You will frequently need to "split terms" to apply it properly. (e.g., to find the min value of x² + 2/x, split it as x² + 1/x + 1/x and apply AM ≥ GM on these 3 terms).
Weighted AM-GM (Advanced Insight): For positive numbers:
( x₁ + x₂ + ... + xₙ ) / n ≥ (x₁ x₂ ... xₙ)^1/n
Used in complex optimization problems.
Evaluating Σ Limits: Master expanding standard series. Questions often combine ∑n and ∑n² inside a limit as n → ∞. Rule of thumb: divide by the highest power of n to find the limit.
Recognizing Mixed Sequences (AGP): You might see an Arithmetico-Geometric Progression where terms look like a, (a+d)r, (a+2d)r² …
Solving trick: Multiply the whole sum S by r, shift it one position to the right, and subtract from the original S to convert it into a pure geometric progression.
S_n Form Recognition: If you are given that S_n = An² + Bn, recognize instantly that the sequence is an AP, and its common difference is 2A.

8. Practice Mock Test

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