Number sequences kya hain?

Whole numbers ke patterns

Maths ke sabse buniyaadi patterns mein whole numbers ke patterns aate hain: 0, 1, 2, 3, 4, …. Inhe padhne waali shaakha ko number theory kehte hain. Number sequence numbers ki aisi list hai jo kisi niyam pe chale, aur number sequences mathematicians ke sabse rochak patterns mein se hain.

Das mukhya sequences

Chapter das zaroori sequences batata hai. Yahan har ek ke agle teen numbers aur niyam hain:

Sequence	Book mein numbers	Agle teen	Niyam
All 1’s	1, 1, 1, 1, 1, 1, 1	1, 1, 1	har number 1
Counting numbers	1, 2, 3, 4, 5, 6, 7	8, 9, 10	1 jodo
Odd numbers	1, 3, 5, 7, 9, 11, 13	15, 17, 19	1 se shuru, 2 jodo
Even numbers	2, 4, 6, 8, 10, 12, 14	16, 18, 20	2 se shuru, 2 jodo
Triangular numbers	1, 3, 6, 10, 15, 21, 28	36, 45, 55	2, phir 3, phir 4, … jodo
Squares	1, 4, 9, 16, 25, 36, 49	64, 81, 100	n × n
Cubes	1, 8, 27, 64, 125, 216	343, 512, 729	n × n × n
Virahānka numbers	1, 2, 3, 5, 8, 13, 21	34, 55, 89	pichhle do numbers jodo
2 ki powers	1, 2, 4, 8, 16, 32, 64	128, 256, 512	2 se guna
3 ki powers	1, 3, 9, 27, 81, 243, 729	2187, 6561, 19683	3 se guna

Niyam kaise dhoondhein

Kisi sequence ka niyam pakadne ke liye kuch sawal poochho:

1. Kya har baar ek hi number juda hai? Odd aur even numbers har baar 2 badhte hain.
2. Kya antar khud badh rahe hain? Triangular numbers 2, 3, 4, 5, … se badhte hain.
3. Kya har number guna ho raha hai? 2 ki powers double hoti hain; 3 ki powers teen guna.
4. Kya har number pichhle numbers se ban raha hai? Har Virahānka number pichhle do ka jod hai.
5. Kya ye number khud se guna hai? Squares 1 × 1, 2 × 2, 3 × 3, … hain; cubes 1 × 1 × 1, 2 × 2 × 2, …

Apna niyam hamesha saare numbers pe parkho, sirf pehle do pe nahi.

Virahānka numbers

1, 2, 3, 5, 8, 13, 21, … mein har number pichhle do ka jod hai: 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, aur aage. (Extra: inhe Fibonacci numbers bhi kehte hain; kavita aur chhand ke Bharatiya vidwaan Virahānka ne inhe Fibonacci se sadiyon pehle bataaya tha.)

2 aur 3 ki powers

2 ki powers har kadam pe double hoti hain: 1, 2, 4, 8, 16, …. Chapter inhe dekhne ka ek sundar tareeka dikhaata hai: ek bindu; line se jude do bindu; 4 konon waala square; 8 konon waala cube; aur 16 ke liye do jude cubes. 3 ki powers har kadam pe teen guna hoti hain: 1, 3, 9, 27, 81, …. Inhe dekhne ka ek tareeka: ek bindu, 3 binduon ki row, 9 binduon ka 3 × 3 square, aur aise 3 squares jama karke 27 ka cube.

Sequences kyun zaroori hain

Number sequences sirf paheliyan nahi hain. Ye aapas mein anokhe tareekon se jude hain, jaise odd numbers jodne se squares milte hain, aur shapes mein bhi dikhte hain, jaise complete graphs ki lines mein. Tasveerein aksar samjhaati hain ki ye patterns kyun hote hain. Iske baare mein padho triangular aur square numbers mein.