MathsHiranyagarbha · Class 11-12
AP vs GP: jab farak jodta hai aur jab guna karta hai
Ek jaisi shuruaat par simple aur compound interest dheere-dheere itna alag kyun hote jaate hain?
Socho pehle… phir dekho jawab
Kyunki simple interest AP banata hai (har saal ek barabar rakam jud-ti hai) aur compound interest GP banata hai (har saal pichhli rakam ka ek hissa aur jud-ta hai). GP ki badhotri samay ke saath tez hoti jaati hai.
Definition: AP mein term mein farak sthir (d) hota hai; GP (Geometric Progression) mein anupaat sthir (r) hota hai: a, ar, ar^2, ... n-wan term a x r^(n - 1). Jod: S(n) = a(r^n - 1)/(r - 1), r alag 1 se. Agar |r| < 1 ho toh anant GP ka jod a/(1 - r) hai: jaise 1 + 1/2 + 1/4 + ... = 2.
Mechanism: maan lo Rs 100 par 10% saalana (yeh sirf ek illustrative example hai). Simple interest: 100, 110, 120, 130 — har saal Rs 10 jud-te hain (AP, d = 10). Compound interest: 100, 110, 121, 133.1 — har saal pichhli rakam ka 1.1 guna (GP, r = 1.1). 3 saal baad compound mein Rs 133.1 aur simple mein Rs 130.
Trade-off: GP ki raftaar lambe samay mein bahut tez hoti hai, par r > 1 tabhi tak rehta hai jab tak asli duniya ki seema (jaise sansaadhan) rok na de. Model sirf sthir d ya sthir r maanta hai; asli cheezein aksar dono se alag chalti hain.
Ghar pe try karo
Teen jawab nikalo: (1) GP 2, 6, 18, ... ka 5wan term; (2) GP 1, 2, 4, 8 ka jod; (3) Rs 100 par 10% compound se 3 saal baad rakam. Phir ek line likho: AP aur GP ka farak kya hai?
Wow!
1 + 1/2 + 1/4 + ... aise anant terms ka jod bhi sirf 2 hota hai!
3 sawaal
- GP 2, 6, 18, ... ka 5wan term kya hai?
Jawab dekho
162. r = 3, isliye 2 x 3^4 = 162.
- 1 + 1/2 + 1/4 + 1/8 + ... ka anant jod kya hai?
Jawab dekho
2. a/(1 - r) = 1/(1 - 1/2) = 2.
- Rs 100 par 10% saalana compound se 3 saal baad kitna hoga (illustrative)?
Jawab dekho
133.1. 100 x 1.1^3 = 133.1.
NCERT se jodo: NCERT Class 11 Maths — sequences and series