Lagnever AI English

Algebra se patterns kaise samjhein?

Short mein jawab

Algebra patterns ko dikhata aur samjhaata hai. Niyam shabdon mein batao, letter-numbers chuno, phir expression likho: A, B, C, A, B, C, … mein nth C 3n pe hai; maachis ke triangles mein step y pe 2y + 1 teeliyan; aur calendar ke 2 × 2 square mein upar-baayein a ho to dono vikarn 2a + 8.

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Pehle pattern, phir expression

Algebraic expressions sirf niyam likhte nahi: ye dikha sakte hain ki koi pattern hamesha kyun chalega. Tareeka hamesha ek hi hai. Rishte ko saral shabdon mein batao ya uska chitra banao, phir badalne waali maatra ki jagah letter-number rakhkar expression likho.

Formula Detective

Ek number machine Y ke upar do sankhyaayein leti hai aur neeche ek deti hai, har baar ek hi kaam karke. Ek machine 5 aur 2 se 8, 8 aur 1 se 15, 9 aur 11 se 7, aur 10 aur 10 se 10 banaati hai. Iska niyam pehli sankhya ka do guna minus doosri hai, 2a − b, isliye 6 aur 4 se 2 × 6 − 4 = 8 milta hai. Doosri machine 5 aur 2 se 5, aur 9 aur 11 se 18 deti hai: ye sankhyaayein jodkar 2 ghataati hai, a + b − 2. Teesri 4 aur 1 se 5, aur 6 aur 0 se 1 deti hai: ye guna karke 1 jodti hai, ab + 1, isliye 10 aur 3 se 31.

Teen number machines unke inputs aur outputs ke saath: 2a − b, a + b − 2 aur ab + 1
Formula Detective

Dohraate designs

Somjit ko saree ki kinaari pe A, B aur C designs dohraate dikhte hain. Design C positions 3, 6, 9, … yaani 3 ke guna-phalon pe hai, isliye uska nth aana 3n pe hai. B hamesha ek jagah pehle, 3n − 1 pe, aur A do jagah pehle, 3n − 2 pe hai. Ulta jaane ke liye position ko 3 se bhaag do: shesh 0 matlab C, 2 matlab B aur 1 matlab A. Isliye position 99 pe C, 122 pe B aur 148 pe A hai.

A, B, C designs dohraate hue, unki positions 3n − 2, 3n − 1 aur 3n, aur pehchaanne waale shesh
A, B, C, A, B, C, …

Calendar ke patterns

November 2024 ke calendar mein 12, 13, 19 aur 20 waale 2 × 2 square ke vikarnon ka jod barabar hai: 12 + 20 = 32 aur 13 + 19 = 32. Kya ye har 2 × 2 square ke liye sach hai, bina ant ke badhaaye calendar mein bhi? Sabko jaanchna sambhav nahi, par algebra faisla kar deta hai. Upar-baayein sankhya ko a kaho. Uske daayein a + 1, neeche a + 7, aur tirchhe neeche a + 8 hai. Vikarnon ke jod a + (a + 8) = 2a + 8 aur (a + 1) + (a + 7) = 2a + 8 hain: har a pe barabar, isliye har square ke liye.

Yahi tareeka doosre aakaaron pe bhi chalta hai. a kendra waale plus aakaar mein sankhyaayein a − 7, a − 1, a, a + 1 aur a + 7 hain, aur inka jod 5a hai, yaani kendra ka paanch guna.

Calendar ka 2 × 2 square a, a + 1, a + 7, a + 8 aur barabar vikarn jod 2a + 8, aur plus aakaar jiska jod 5a hai
Calendar ke patterns

Maachis ke patterns

Maachis ki teeliyon se bane triangles ki line mein step 1 ke liye 3, step 2 ke liye 5, step 3 ke liye 7 aur step 4 ke liye 9 teeliyan lagti hain: har step 2 jodta hai. Step y pehli 3 teeliyon mein y − 1 baar 2 jodta hai, isliye 3 + 2(y − 1) teeliyan, jo simplify hokar 2y + 1 banta hai. Pehle 3 ko 1 + 2 dekhne se seedha 2y + 1 milta hai. Doosre tareeke se ginein to upar-neeche y leti teeliyan aur y + 1 tirchhi teeliyan hain, jo phir 2y + 1 banti hain. Isliye step 33 ke liye 67, step 84 ke liye 169 aur step 108 ke liye 217 teeliyan.

Step 1 se 4 ke maachis ke triangles 3, 5, 7 aur 9 teeliyon ke saath, aur formula 2y + 1
Maachis ke triangles

Ye kyun zaroori hai

Ganitagya kisi pattern pe tab tak bharosa nahi karte jab tak use samjhaa na sakein, aur algebra isi ka tareeka hai. Expressions likhne ki buniyaad letter-numbers kya hain mein, aur in proofs mein laga simplify karna expressions simplify kaise karein mein hai. Traffic signal, rassi aur number grid jaise aur patterns Chapter 4 ke NCERT solutions mein hal kiye gaye hain; poore Chapter 4 ke notes bhi dekho.

Practice karo

  1. Ek number machine 4 aur 1 se 5, 6 aur 0 se 1, aur 3 aur 2 se 7 deti hai. Formula kya hai?

    1. (a) a + b
    2. (b) ab + 1
    3. (c) 2a − b
    4. (d) a + b − 2
    Jawab dekho

    Jawab: (b) ab + 1

  2. A, B, C, A, B, C, … mein nth B kahan hai?

    1. (a) 3n
    2. (b) 3n − 1
    3. (c) 3n − 2
    4. (d) n + 1
    Jawab dekho

    Jawab: (b) 3n − 1

  3. Triangle pattern (2y + 1) ke step 33 ke liye kitni teeliyan chahiye?

    1. (a) 66
    2. (b) 67
    3. (c) 99
    4. (d) 35
    Jawab dekho

    Jawab: (b) 67

  4. Calendar mein plus aakaar ka kendra a hai. Dikhao ki paanchon sankhyaon ka jod kendra ka 5 guna hai.

    Jawab dekho

    Jawab: Upar aur neeche ki sankhyaayein a − 7 aur a + 7, aur baayein-daayein a − 1 aur a + 1 hain. Jod (a − 7) + (a − 1) + a + (a + 1) + (a + 7) = 5a hai, kyunki sankhyaayein kat jaati hain.

  5. A, B, C, A, B, C, … mein position 148 pe kaunsa design hai?

    Jawab dekho

    Jawab: 148 = 3 × 49 + 1, yaani 3 se bhaag dene pe shesh 1. Shesh 1 waali positions pe design A hai, isliye position 148 pe A hai.

Students ke sawal

Patterns ko algebra se kaise dikhaate hain?

Pehle pattern ko shabdon mein batao, phir badalne waali maatra ke liye letter-number chuno, aur niyam ko expression ki tarah likho. A, B aur C dohraane waali saree ki kinaari mein C, 3 ke guna-phalon pe aata hai, isliye nth C position 3n pe hai; B usse theek pehle 3n − 1 pe, aur A 3n − 2 pe.

Calendar ke 2 × 2 square ke vikarnon ka jod hamesha barabar kyun hota hai?

Upar-baayein taareekh a ho to uske daayein a + 1, neeche a + 7 aur tirchhe neeche a + 8 hai. Vikarnon ke jod a + (a + 8) = 2a + 8 aur (a + 1) + (a + 7) = 2a + 8 hain. Ye a ki har value pe barabar hain, isliye har 2 × 2 square ke liye, bina ant ke badhaaye calendar mein bhi.

Maachis ke pattern ka formula kaise nikaalein?

Pehle kuch steps ki teeliyan gino aur dekho ve kaise badhti hain. Line mein triangles ke liye 3, 5, 7, 9, …: 3 se shuru, har naye step pe 2 jodo. Step y pe y − 1 baar jodna hota hai, isliye kul 3 + 2(y − 1), jo simplify hokar 2y + 1 banta hai. Leti teeliyan (y) aur tirchhi teeliyan (y + 1) gin-ne se bhi yahi jawab aata hai.

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