📈 Series and Progressions (AP & GP)

Day 3 | ⏱️ Part of 3-Hour Advanced Quant Block | TCS Interview Prep

📌 Subtopics

• Arithmetic Progression (AP): nth term, sum of n terms, common difference
• Geometric Progression (GP): nth term, sum of n terms, common ratio, infinite GP
• Harmonic Progression (HP): Reciprocal of AP
• Mixed series problems

📖 Comprehensive Notes

1. Arithmetic Progression (AP)

An AP is a sequence of numbers in which the difference between consecutive terms is constant, called the common difference ($d$).

• $n$-th Term ($a_n$ or $T_n$): $a_n = a + (n - 1)d$ where $a$ is the first term.
• Sum of first $n$ terms ($S_n$): $S_n = \frac{n}{2} [2a + (n-1)d] = \frac{n}{2} [a + l]$ where $l$ is the last term.
• Arithmetic Mean (AM): If $a, b, c$ are in AP, then $b = \frac{a+c}{2}$.

Example: Find the sum of the first $20$ terms of the AP: $5, 8, 11, 14, \dots$ $a = 5$, $d = 3$, $n = 20$. $S_{20} = \frac{20}{2} [2(5) + (20-1)3] = 10 [10 + 57] = 10 \times 67 = 670$.

2. Geometric Progression (GP)

A GP is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$).

• $n$-th Term ($a_n$ or $T_n$): $a_n = a \cdot r^{n-1}$
• Sum of first $n$ terms ($S_n$): $S_n = \frac{a(r^n - 1)}{r - 1} \text{ (if } r > 1) \quad \text{or} \quad S_n = \frac{a(1 - r^n)}{1 - r} \text{ (if } r < 1)$
• Sum of Infinite GP ($S_\infty$): Only when the absolute value of the common ratio is less than $1$ ($|r| < 1$): $S_\infty = \frac{a}{1 - r}$
• Geometric Mean (GM): If $a, b, c$ are in GP, then $b^2 = ac \implies b = \sqrt{ac}$.

Example: Find the sum of the infinite geometric series: $1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots$ $a = 1$, $r = 1/2$ (since $|r| < 1$, infinite sum exists). $S_\infty = \frac{1}{1 - 1/2} = \frac{1}{1/2} = 2$.

3. Harmonic Progression (HP)

A sequence of numbers $a_1, a_2, a_3, \dots$ is in HP if their reciprocals $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \dots$ are in AP.

• There is no direct formula for the sum of an HP. Convert the terms to AP, perform calculations, and take the reciprocal at the end.
• Harmonic Mean (HM): If $a, b, c$ are in HP, then: $b = \frac{2ac}{a+c}$
• Relation between AM, GM, HM: For positive real numbers: $\text{AM} \ge \text{GM} \ge \text{HM}$ $\text{GM}^2 = \text{AM} \times \text{HM}$

Example: Find the 10th term of the HP: $\frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{9}, \dots$ Reciprocals in AP: $3, 5, 7, 9, \dots$ First term of AP $a = 3$, common difference $d = 2$. 10th term of AP $= 3 + (10-1)2 = 3 + 18 = 21$. Therefore, 10th term of HP $= \frac{1}{21}$.

🌐 Learning Resources

✅ Practice Links

📝 Key Formulas

ARITHMETIC PROGRESSION (AP):
  nth term:     aₙ = a + (n-1)d
  Sum of n:     Sₙ = n/2 × [2a + (n-1)d]  OR  Sₙ = n/2 × (first + last)

GEOMETRIC PROGRESSION (GP):
  nth term:     aₙ = a × r^(n-1)
  Sum of n:     Sₙ = a(rⁿ - 1)/(r - 1)     [r > 1]
                Sₙ = a(1 - rⁿ)/(1 - r)     [r < 1]
  Sum to ∞:     S∞ = a/(1 - r)              [|r| < 1]

HARMONIC PROGRESSION (HP):
  If a, b, c are in HP → 1/a, 1/b, 1/c are in AP
  Harmonic Mean = 2ab/(a+b)

🔗 Related Topics

• Patterns and Series — Reasoning-style pattern-based sequence/coding logic (distinct but related)
• Equations — Solving systems of equations for sequence parameters like a or d
• Dynamic Programming — Fibonacci and sequence sum optimization problems in coding
• Arrays and Hashing — Sequence generation algorithms
• Greedy Number Theory and Bit Manipulation — Number sequences and modular progression logic