Sequences & Series 📈

Patterns dhundhna — numbers ki duniya mein order nikalna. AP, GP, aur bhi bahut kuch!

💰 Bank mein Paisa

Maan lo tumhare paas ₹1000 hain. Ek bank Simple Interest deta hai ₹100/year (fixed amount — yeh AP hai!). Doosra bank Compound Interest deta hai — pehle saal ₹100, doosre saal ₹110, teesre saal ₹121 (ratio fixed — yeh GP hai!). GP mein growth exponential hai — isliye long-term investment mein hamesha compound interest better hota hai. Yahi sequences ka real world magic hai!

1. Sequence aur Series — Difference

Sequence: Numbers ka ek ordered list jisme ek pattern hota hai — har term aage wale se connected hai. Jaise: 1, 4, 9, 16, 25, ... (perfect squares)

Series: Sequence ke terms ko + se jodna. Jaise: 1 + 4 + 9 + 16 + 25 + ... Series ka sum 'Sₙ' se denote karte hain.

General term (aₙ): nth term ka formula — isse koi bhi term directly nikaal sakte hain.

2. Arithmetic Progression (AP)

Jab har consecutive terms ka difference constant ho — ise common difference 'd' kehte hain. Yeh common difference add karte raho, AP banata rehta hai.

AP ke Basic Formulas

First term = a, Common difference = d = a₂−a₁ = a₃−a₂ = ...

nth term: aₙ = a + (n−1)d
Sum of n terms: Sₙ = n/2 [2a + (n−1)d] = n/2 [a + l]

Jahan l = last term = a + (n−1)d

AP: 2, 5, 8, 11, 14 ... (d=3)

2 5 8 11 14 +3 +3 +3 +3

✅ AP: 3, 7, 11, 15, ... ka 20th term aur S₂₀ nikalo

a = 3, d = 7−3 = 4, n = 20
a₂₀ = 3 + (20−1)×4 = 3 + 76 = 79
S₂₀ = 20/2 × [2(3) + 19×4] = 10 × [6 + 76] = 10 × 82 = 820

3. Geometric Progression (GP)

Jab har consecutive terms ka ratio constant ho — ise common ratio 'r' kehte hain. Pehle term ko r se multiply karte raho, GP banta rehta hai.

GP ke Basic Formulas

First term = a, Common ratio = r = a₂/a₁ = a₃/a₂ = ...

nth term: aₙ = a · rⁿ⁻¹
Sum (r ≠ 1): Sₙ = a(rⁿ−1)/(r−1) jab r > 1
Infinite GP (|r|<1): S∞ = a/(1−r)

GP: 2, 6, 18, 54, 162 ... (r=3)

2 6 18 54 162 ×3 ×3 ×3 ×3

✅ GP: 1, 2, 4, 8, ... ka 8th term aur S₆ nikalo

a = 1, r = 2, n = 8
a₈ = 1 × 2⁷ = 128
S₆ = 1×(2⁶−1)/(2−1) = (64−1)/1 = 63

✅ Infinite GP: 1 + 1/2 + 1/4 + 1/8 + ... ka sum nikalo

a = 1, r = 1/2 (|r| < 1, toh infinite sum exist karta hai)
S∞ = a/(1−r) = 1/(1−1/2) = 1/(1/2) = 2

Interesting: Infinite terms add karo → exact finite answer milta hai jab |r|<1!

4. AP vs GP — Comparison

PropertyAPGP
Common featureCommon difference d (add)Common ratio r (multiply)
nth terma + (n−1)da·rⁿ⁻¹
Sum of n termsn/2[2a+(n−1)d]a(rⁿ−1)/(r−1)
Nature of growthLinear (straight line)Exponential (curve)
Example2,5,8,11,14 ... (d=3)2,6,18,54 ... (r=3)

5. Special Series — Σ Formulas (Important!)

Yeh teen summation formulas bahut commonly exam mein aate hain — zaroor yaad karo:

Σk = 1+2+3+...+n = n(n+1)/2
Σk² = 1²+2²+3²+...+n² = n(n+1)(2n+1)/6
Σk³ = 1³+2³+...+n³ = [n(n+1)/2]²

✅ 1² + 2² + 3² + ... + 10² nikalo

n=10 → Σk² = 10(11)(21)/6 = 2310/6 = 385

6. AM aur GM

Arithmetic Mean (AM): Do numbers a aur b ke beech ka term jo AP banaye = (a+b)/2

Geometric Mean (GM): Do numbers a aur b ke beech ka term jo GP banaye = √(ab)

AM ≥ GM hamesha (equality jab a = b)

Yeh inequality bahut powerful hai — optimization problems mein aata hai!

7. Practice Questions

Q1. AP: 5, 11, 17, 23, ... ka 15th term aur S₁₅ nikalo.

Solution

a=5, d=6. a₁₅=5+14×6=89. S₁₅=15/2×[10+84]=15/2×94=705

Q2. GP: 3, 9, 27, ... ka 6th term nikalo.

Solution

a=3, r=3. a₆=3×3⁵=3×243=729

Q3. Ek AP mein 5th term = 24 aur 10th term = 44. AP aur S₁₀ nikalo.

Solution

a₅=a+4d=24, a₁₀=a+9d=44. Subtract: 5d=20, d=4. a=8. AP: 8,12,16... S₁₀=10/2[16+36]=10/2×52=260

Q4. 1+2+3+...+50 ka sum kya hai?

Solution

n(n+1)/2 = 50×51/2 = 1275

Q5. Infinite GP: 8, 4, 2, 1, ... ka sum nikalo.

Solution

a=8, r=1/2. S∞=8/(1−1/2)=8/(1/2)=16

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