MathsPushan · Class 9-10
Arithmetic Progression: 100wan term bina saare likhe
1 + 2 + 3 + ... + 100 ka jod ek hi minute mein kaise nikal sakte ho?
Socho pehle… phir dekho jawab
Pairs banakar! Pehla aur aakhri term jodo, doosra aur doosra-aakhri jodo — har pair ka jod 101 aata hai, aur aise 50 pairs hain: 50 x 101 = 5050.
Mechanism: AP (Arithmetic Progression) wo sequence hai jisme har term pichhle se ek sthir sankhya d (common difference) se badhta ya ghat-ta hai. Pehla term a ho toh n-wan term a(n) = a + (n - 1)d. Jaise 5, 8, 11, ... mein a = 5, d = 3, isliye 10wan term = 5 + 9 x 3 = 32.
Number yaad rakho: pehle n terms ka jod S(n) = n/2 x [2a + (n - 1)d]. Agar pehla term a aur aakhri term l ho toh S(n) = n/2 x (a + l). 1 se n tak ke natural numbers ka jod n(n + 1)/2 hota hai, isliye 1 se 100 tak ka jod 100 x 101 / 2 = 5050.
So what: AP har jagah sthir badhotri dikhata hai — jaise har mahine ek barabar rakam bachana. Par yaad rakho, AP tabhi sahi hai jab farak sach mein sthir ho; asli zindagi mein kai cheezein sthir farak se nahi badhtin.
Ghar pe try karo
AP 2, 4, 6, ... ke pehle 10 terms ka jod S(n) formula se nikalo, phir ek-ek jodkar jaanch lo. Dono jawab milne chahiye.
Wow!
AP ke jod ka trick pairs banana hai — pehla + aakhri, doosra + doosra-aakhri, sab ka jod ek jaisa!
3 sawaal
- AP 5, 8, 11, ... ka 10wan term kya hai?
Jawab dekho
32. a(10) = 5 + 9 x 3 = 32.
- 1 + 2 + 3 + ... + 100 ka jod kya hai?
Jawab dekho
5050. n(n + 1)/2 = 100 x 101 / 2 = 5050.
- AP mein har term aur uske pichhle term ka farak kaisa hota hai?
Jawab dekho
Hamesha sthir (common difference). AP ki pehchaan hi sthir common difference d hai.
NCERT se jodo: NCERT Class 10 Maths — arithmetic progressions