Triangular aur square numbers kya hain?
Short mein jawab
Triangular numbers, 1, 3, 6, 10, 15, …, utne bindu hain jo triangle mein jamaaye ja sakein; square numbers, 1, 4, 9, 16, 25, …, utne bindu jo square grid bhar dein. 1 se odd numbers jodne pe squares milte hain, aur lagaataar do triangular numbers ka jod bhi square hota hai. 36 triangular bhi hai aur square bhi.
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Numbers jo banaaye ja sakte hain
Kuch numbers ka naam unke binduon ki shape se aata hai. Triangular numbers, 1, 3, 6, 10, 15, …, utne bindu hain jo 1, 2, 3, … binduon ki rows waale triangle mein jamaaye ja sakein. Isliye har triangular number 1 + 2 + 3 + … kisi number tak ka jod hai: 10 = 1 + 2 + 3 + 4. Square numbers, 1, 4, 9, 16, 25, …, utne bindu hain jo square grid bhar dein: 1 × 1, 2 × 2, 3 × 3, aur aage.
36: triangle bhi, square bhi
Ek hi number alag roop le sakta hai. 36 bindu 6 by 6 square mein jamaaye ja sakte hain, aur 1, 2, 3, 4, 5, 6, 7 aur 8 binduon ki rows waale triangle mein bhi, kyunki 1 + 2 + … + 8 = 36. Isliye 36 square number bhi hai aur triangular number bhi.
Odd numbers jodne se squares
Dekho jab hum odd numbers jodte hain to kya hota hai: 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16, 1 + 3 + 5 + 7 + 9 = 25. Jawab squares hain. Ye hamesha kyun hota hai? Binduon ke square grid ko L-shaped parton mein baanto: kone ka bindu, phir uske chaaron taraf 3 bindu, phir 5, phir 7, aur aage. Har naya odd number square ke chaaron taraf ek aur L lapet ke agla bada square bana deta hai. Ye kisi bhi size ke square ke liye hota hai, isliye pattern hamesha chalta rehta hai. Pehle 10 odd numbers ka jod 10 × 10 = 100, aur pehle 100 odd numbers ka 100 × 100 = 10,000 hai.
Upar aur neeche jodna
Counting numbers ko upar aur phir neeche jodo:
- 1 + 2 + 1 = 4
- 1 + 2 + 3 + 2 + 1 = 9
- 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16
Phir se squares! Tasveer samjhaati hai: 5 by 5 binduon ke square ki tirchi rows mein 1, 2, 3, 4, 5, 4, 3, 2 aur 1 bindu hain. Isliye 1 + 2 + 3 + … + 99 + 100 + 99 + … + 3 + 2 + 1 = 100 × 100 = 10,000.
Do triangles se ek square
Binduon ke square ko seedhi (staircase) ki tarah kaato to do triangles milte hain, ek doosre se ek kadam chhota. Isliye lagaataar do triangular numbers ka jod hamesha square hota hai: 1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, 10 + 15 = 25.
Sequences ke aur rishte
| Shuruaat | Kya karo | Kya milta hai |
|---|---|---|
| All 1’s | jodo: 1, 1 + 1, 1 + 1 + 1, … | counting numbers |
| Counting numbers | jodo: 1, 1 + 2, 1 + 2 + 3, … | triangular numbers |
| Triangular numbers | 6 se guna karke 1 jodo | hexagonal numbers 7, 19, 37, 61, … |
| Hexagonal numbers | jodo: 1, 1 + 7, 1 + 7 + 19, … | cubes 1, 8, 27, 64, … |
Shapes se bane patterns ke liye padho shapes mein patterns.
Practice karo
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Saatvaan triangular number kya hai?
Jawab dekho
Jawab: (b) 28 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.
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Pehle 10 odd numbers ka jod kya hai?
Jawab dekho
Jawab: (c) 100
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1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 kitna hai?
Jawab dekho
Jawab: (b) 36
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Kaunse do lagaataar triangular numbers ka jod 49 hai?
Jawab dekho
Jawab: 21 aur 28: 21 + 28 = 49 = 7 × 7.
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Dikhao ki 36 triangular bhi hai aur square bhi.
Jawab dekho
Jawab: 36 = 6 × 6, isliye 36 bindu 6 by 6 square bharte hain. Saath hi 36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8, isliye 36 bindu 1 se 8 binduon ki rows waala triangle banaate hain.
Students ke sawal
1, 3, 6, 10, 15 ko triangular numbers kyun kehte hain?
Kyunki itne bindu triangle mein jamaaye ja sakte hain: 1, phir 1 aur 2, phir 1, 2 aur 3 ki rows, aur aage. Har triangular number 1 + 2 + 3 + … kisi number tak ka jod hai.
1, 4, 9, 16, 25 ko square numbers kyun kehte hain?
Kyunki itne bindu square grid bhar dete hain: 1 × 1, 2 × 2, 3 × 3, 4 × 4, 5 × 5.
Odd numbers jodne se square numbers kyun milte hain?
Binduon ka square grid 1, 3, 5, 7, … binduon ki L-shaped parton mein baanta ja sakta hai. Har naya odd number square ke chaaron taraf ek aur L jodkar agla bada square bana deta hai.
Pehle 100 odd numbers ka jod kya hai?
100 × 100 = 10,000, kyunki pehle n odd numbers ka jod hamesha n × n hota hai.
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