3.3.2
GP ek aisi sequence hai jisme aap har step pe same number se multiply karte ho (add nahi, jaise AP mein hota hai).
KYA HAI: har term, pichli term ko ek fixed ==common ratio r KYUN ZAROORI HAI: baar baar multiplication exponential hoti hai — yeh compound interest, population growth, radioactive decay, aur geometry (areas ka aadha hona) model karta hai. Nature "add" karne se kahin zyada "multiply" karta hai.KAISE PEHCHANO: kisi bhi term ko uske pehle wali term se divide karo. Agar woh ratio constant hai, toh yeh GP hai.
Definition Geometric progressionEk sequence a 1 , a 2 , a 3 , … GP hai agar consecutive terms ka ratio constant ho:
a n a n + 1 = r ( same for all n ) , r = 0 , a 1 = 0.
a a 1 first term r common ratio Example: 3 , 6 , 12 , 24 , … a = 3 r = 2
a = 0 r = 0 a = 0 0 r = 0 0
Intuition Pattern kyun haiHar nayi term paane ke liye tum r n ( n − 1 ) r n ( n − 1 ) gaps hote hain.
Derivation (Feynman style, ek baar mein ek multiplication):
a 1 a 2 a 3 a 4 a n = a = a 1 ⋅ r = a r = a 2 ⋅ r = a r 2 = a 3 ⋅ r = a r 3 ⋮ = a r n − 1 Exponent n n − 1 Pehli term mein zero multiplications hote hain (r 0 = 1
Intuition Clever trick kyun exist karta haia + a r + a r 2 + … poore sum ko r shift ho jaati hai. Dono lines subtract karne par almost sab cancel ho jaata hai (ek telescoping cascade), sirf ends bach jaate hain.
Derivation:
Maano
S_n = a + ar + ar^2 + \cdots + ar^{\,n-1}. \tag{1}
Har term ko r rS_n = ar + ar^2 + ar^3 + \cdots + ar^{\,n}. \tag{2}
r terms ek jagah shift hokar reproduce karta hai, taaki wo cancel ho jaayein.
(1) mein se (2) subtract karo:
S n − r S n = a − a r n Beech ka hissa kyun gayab ho jaata hai? Har interior term a r k dono lines mein appear karta hai, isliye wo cancel ho jaati hai. Sirf (1) ki pehli term aur (2) ki aakhiri term bachti hai.
Factor karo:
S n ( 1 − r ) = a ( 1 − r n ) ( 1 − r ) lekin sirf tab jab r = 1
S n = 1 − r a ( 1 − r n ) = r − 1 a ( r n − 1 ) Do forms kyun? Ye algebraically identical hain (upar aur neeche dono ko − 1
Worked example Example 1 — ek term dhundhoGP: 5 , 10 , 20 , …
Step: a = 5 , r = 10/5 = 2 Kyun? Consecutive terms ka ratio r Step: a 7 = a r 6 = 5 ⋅ 2 6 Exponent 6 kyun? n − 1 = 7 − 1 = 5 ⋅ 64 = 320
Worked example Example 2 — sum jabr > 1 3 , 6 , 12 , …
a = 3 , r = 2 , n = 6 S n = r − 1 a ( r n − 1 ) Yeh form kyun? r > 1 S 6 = 2 − 1 3 ( 2 6 − 1 ) = 3 ( 64 − 1 ) = 3 ⋅ 63 = 189
Worked example Example 3 — fractional ratio (r < 1 8 , 4 , 2 , 1
a = 8 , r = 2 1 , n = 4 S n = 1 − r a ( 1 − r n ) Yeh form kyun? r < 1 1 − r > 0 S 4 = 1 − 2 1 8 ( 1 − ( 2 1 ) 4 ) = 2 1 8 ( 1 − 16 1 ) = 16 ⋅ 16 15 = 15 Check: 8 + 4 + 2 + 1 = 15
Worked example Example 4 —n 2 , 6 , 18 , … 486
Forecast: terms ×3 se badhti hain, toh n
Solve karo a r n − 1 = 486 ⇒ 2 ⋅ 3 n − 1 = 486 ⇒ 3 n − 1 = 243 = 3 5 ⇒ n − 1 = 5 ⇒ n = 6
Common mistake nth term ke liyer n − 1 r n Kyun sahi lagta hai: "n n Fix: Pehli term ki power 0 a 1 a r 1 − 1 = a r 0 = a r n a r = a
r = 1 Kyun sahi lagta hai: formula "hamesha kaam karta hai". Fix: r = 1 1 − r = 0 r = 1 a S n = na
Common mistake Do sum forms mix karne se galat signKyun sahi lagta hai: tumne ek form yaad ki aur aisa case mein daali jisme woh suit nahi karti.
Fix: r > 1 r − 1 a ( r n − 1 ) r < 1 1 − r a ( 1 − r n )
r Kyun sahi lagta hai: AP mein d Fix: GP mein division use hoti hai: r = a n + 1 / a n a n + 1 − a n
Recall Feynman: ek 12-saal ke bacche ko samjhaoEk magic photocopier imagine karo jo hamesha double size ki copies banata hai (ya aadha, ya triple). Size 3 ka ek sticker se shuru karo. Andar daalo: 6, phir 12, phir 24 — har baar woh same number se multiply karta hai. Yahi GP hai! Kuch copies ke baad size jaanne ke liye, tum sirf us magic number ko utni baar khud se multiply karte ho jitni baar copy bani — stickers ki sankhya se ek kam, kyunki pehle sticker ki zero baar copy hui thi. Aur agar saare stickers ka total area chahiye, toh ek shortcut hai: unhe ek-ek karke add karne ki jagah, poori pile ko copy karo, ek jagah shift karke rakho, aur subtract karo — almost sab cancel ho jaata hai aur sirf pehla aur aakhiri hissa bachta hai. Neat!
"First term times ratio, ONE less power" → a r n − 1 "upar aur neeche dono mein ( 1 − r ) n r n → 1 − r a ( 1 − r n )
Arithmetic Progression (AP) — GP wahan multiply karta hai jahan AP add karta hai; S n reverse trick se wahan derive hota hai, yahan shifting se.Sum of infinite GP — jab ∣ r ∣ < 1 r n → 0 S ∞ = 1 − r a Geometric Mean — 3-term GP ki middle term: b = a c Exponential functions — GP, a r x Compound interest — amounts ek GP form karte hain jisme r = 1 + 100 i Logarithms — r n − 1 = k n
GP kya define karta hai? Consecutive terms ka ratio constant hota hai: a n + 1 / a n = r
First term a r a n = a r n − 1
Exponent n n − 1 Pehli term r r 0 = 1 n ( n − 1 )
GP ke pehle n r = 1 S n = 1 − r a ( 1 − r n ) = r − 1 a ( r n − 1 )
GP sum kaunsi trick se derive hota hai? S n r
r = 1 S n S n = na a 1 − r = 0
Common ratio r Ek term ko pichli term se divide karo, subtract nahi.
Formula se 8 + 4 + 2 + 1 a = 8 , r = 2 1 , n = 4 1/2 8 ( 1 − 1/16 ) = 15
Infinite GP sum kab exist karta hai aur kiske barabar hota hai? Jab ∣ r ∣ < 1 S ∞ = 1 − r a
Constant ratio a_n+1 / a_n = r
S_n = a + ar + ... + ar^n-1
S_n = a 1-r^n / 1-r, r != 1
Compound interest, growth, decay
