Sequences & Series — Applied Maths 11

📏 1. Arithmetic Progression (AP)

Jisme har term pichle term se ek fixed amount (common difference d) se badta ya ghatta hai.

Pattern: a, a+d, a+2d, a+3d, ...

nth term: aₙ = a + (n−1)d Sum of n terms: Sₙ = n/2 × [2a + (n−1)d] = n/2 × (first term + last term) = n/2 × (a + l) Jahan: a = first term, d = common difference, n = number of terms l = last term = a + (n−1)d

Example 1: nth Term

AP: 7, 11, 15, 19,... ka 20th term kya hoga?

a=7, d=4, n=20: a₂₀ = 7 + (20−1)×4 = 7 + 76 = 83

Example 2: Sum of AP

1 se 100 tak ke sabhi natural numbers ka sum (yahi Gauss ka trick hai!)

a=1, d=1, n=100, l=100

S₁₀₀ = 100/2 × (1+100) = 50 × 101 = 5050

Example 3: Find n given Sum

AP: 3, 5, 7,... ka sum 120 hai. n terms kitne hain?

a=3, d=2: Sₙ = n/2[6+2(n−1)] = n/2(2n+4) = n(n+2) = 120

n² + 2n − 120 = 0 → (n+12)(n−10) = 0 → n = 10

AM (Arithmetic Mean)

a aur b ke beech n AM insert karna: Common difference d = (b − a)/(n + 1) Single AM between a and b = (a + b)/2 Example: 5 aur 23 ke beech 3 AMs: d = (23−5)/4 = 18/4 = 4.5 Means: 9.5, 14, 18.5

📊 2. Geometric Progression (GP)

Jisme har term pichle term se ek fixed number (common ratio r) se multiply hota hai.

Pattern: a, ar, ar², ar³, ...

nth term: aₙ = a × rⁿ⁻¹ Sum of n terms: Sₙ = a(rⁿ − 1)/(r − 1) [jab r ≠ 1] Sₙ = na [jab r = 1] Sum of infinite GP (|r| < 1): S∞ = a / (1 − r)

Example 1: GP nth term

GP: 3, 6, 12, 24,... 8th term kya hai?

a=3, r=2: a₈ = 3 × 2⁷ = 3 × 128 = 384

Example 2: Sum of GP

GP: 1, 2, 4, 8,... ka sum pehle 10 terms mein.

a=1, r=2: S₁₀ = 1×(2¹⁰−1)/(2−1) = (1024−1)/1 = 1023

Example 3: Infinite GP

1 + 1/2 + 1/4 + 1/8 + ... ka sum (infinite terms).

a=1, r=1/2 (|r|<1): S∞ = 1/(1−1/2) = 1/(1/2) = 2

GM (Geometric Mean)

Single GM between a and b = √(ab) n GMs between a and b: Common ratio r = (b/a)^(1/(n+1)) Example: 3 aur 48 ke beech GM = √(3×48) = √144 = 12

⚖️ 3. AM aur GM ka Relationship

For any two positive numbers a and b: AM = (a + b)/2 GM = √(ab) IMPORTANT: AM ≥ GM (hamesha, equality tabhi jab a = b) AM − GM ≥ 0 (a+b)/2 ≥ √(ab) (a+b)² ≥ 4ab

Application: Minimum Value Problem

x > 0 ke liye, x + 1/x ka minimum value nikalo.

AM ≥ GM apply karo: (x + 1/x)/2 ≥ √(x × 1/x) = √1 = 1

x + 1/x ≥ 2. Minimum value = 2 (jab x = 1)

	AP	GP
Pattern	Add/subtract	Multiply/divide
Common diff/ratio	d = aₙ₊₁ − aₙ	r = aₙ₊₁/aₙ
nth term	a+(n−1)d	a·rⁿ⁻¹
Sum n terms	n/2[2a+(n−1)d]	a(rⁿ−1)/(r−1)
Mean	AM = (a+b)/2	GM = √(ab)

🔢 Sequences & Series

📏 1. Arithmetic Progression (AP)

AM (Arithmetic Mean)

📊 2. Geometric Progression (GP)

GM (Geometric Mean)

⚖️ 3. AM aur GM ka Relationship