Hello friend !! Aaj hum aapko class 10 ka ek new chapter padhane jaa rahe hai. Jiska naam hai ,"Arithmetic Progression". Isse hindi me "समांतर श्रेढ़ी (samaantar shreni)" kahte hai. Issko short form me "A. P." bhi kaha jaata hai. "A. P." ka full form ya matlab "Arithmetic Progression" hota hai.

Samaantar shreni ke iss chapter ko aaj hum full detail aur ekdam Basic se padhne jaa rahe hai. Hume puri ummid hai ki, iss se aapko puri help milegi. To chaliye bina late kiye huye aaj ka topic start karte hai-:

Figure-1

Aaj hum jo padhne waale hai, wo kuch iss prakar se hai-:

❒ Samantar shreni (Arithmetic Progression) kya hota hai?
❒ Samantar shreni ke Examples.
❒ Samantar shreni ka Saarv Antar (Common Difference) kya hota hai ?
❒ Samantar shreni ka vyapak roop (general formula) kya hota hai ?
❒ Samantar shreni ke Basic Question and Solution

• Samantar shreni (Arithmetic Progression) kya hota hai?

Samantar shreni ek sidhi rekha (straight line) me likhi aisi sankhyao ka samuh (group) hota hai, "jisme pahli sankhya (first number) ke alawa baaki bache har sankhya apne se pahle aane waale sankhya me ek fix number ko jodane (add) karne par milte hai."

Jaise-: 5, 10, 15, 20, .... ek Samantar shreni hai. Isme 10 apne se pahle pad 5 me 5 add (5+5 = 10) karne par milte hai, 15 apne se pahle pad 10 me 5 add (10+5 = 15) karne par milte hai aur bhi sabhi number issi tarah milta hai. Isliye ye ek Samantar shreni ka example hai.

• Samantar shreni ke Examples -:

(i). 2, 4, 6, 8, ...........

(ii). 31, 28, 25, 22, ..........

(iii). -3, -7 -11, -15, ...........

(iv). -45, -40, -35, -30, ..........

Inn examples me diye huye sabhi numbers samantar shreni ka "pad (term)" kahalate hai.

Upar diye gaye sabhi examples (i), (ii), (iii) aur (iv) samantar shreni ke examples hai, kyuki aap dekh sakte hai ki,

Example (i) me 2 add karne par next (agala) number milta hai.

Example (ii) me -2 add karne par next (agala) number milta hai.

Example (iii) me -4 add karne par next (agala) number milta hai.

Example (iv) me 5 add karne par next (agala) number milta hai.

Samantar shreni ke first term ko a₁, second term ko a₂, third term ko a₃ ... n^th term ko a_n se denote kare toh

a₁, a₂, a₃ . . . a_n ko samantar shreni kahte hai.

Ab hum baat karte hai saarv antar (Common difference) ke baare me,

• Samantar Shreni ka Saarv antar (common difference) kya hota hai?

Samantar shreni ka saarv antar (common difference), "wo fixed number hota hai, jisko hum samantar shreni ke kisi bhi number me add (+) karke hum uss Samantar shreni ka agala (next) number prapt kar sakte hai."

Ye saarv antar, positive(+) number, negative number (-) yaa zero (0) bhi ho sakta hai. Saarv antar ko "d" se prakat (denote) karte hai.

• Kisi samantar shreni ka Saarv antar (common difference) "d" kaise gyaat karte hai ?

Samantar shreni ka common difference "d" gyaat karne ke liye samantar shreni ke kisi bhi number me se uss number ke pahle waale number ko subtract karte hai.

Means a₁, a₂, a₃ . . . a_n samantar shreni ho toh common difference "d" hum iss prakar se gyaat kar sakte hai-:

d = a₃ - a₂ = a₂ - a₁ = . . . = a_n - a_{n - 1}

• Samantar shreni ka vyapak roop (general formula) kya hota hai?

Samantar shreni ka vyapak roop (general formula) ko hum iss prakar se likhte hai-:

a, a+d, a+2d, a+3d, . . .

Iss vyaapak roop me pahla pad (first term) = a aur Saarv antar (Common difference) = d hai. Isse Samantar shreni ka vyapak roop kahte hai.

• Samantar shreni ke Basic Question and Solution

Ab hum samantar shreni se related kuch Basic Question ko solve karna sikhne jaa rahe hai.

Question-1. Yadi pratham pad (first term) a = 2 aur saarv antar d = 4 ho toh Samantar Shreni gyaat kijiye.

Solution-: Diya hai: a = 2 aur d = 4.

Samantar shreni ke vyaapak rup ka prayog karne par :

⇒ a, a+d, a+2d, a+3d, a+4d, . . .

a = 2, a+d = 2+4 = 6,

a+2d = 2+2×4 = 10, a+3d = 2+3×4 = 14

Isliye, Samantar shreni = 2, 6, 10, 14, . . . Ans.

Question-2. Yadi pratham pad (first term) a = -1/3 aur saarv antar d = 1/2 ho toh Samantar Shreni gyaat kijiye.

Solution-: Diya hai: a = -1/3 aur d = 1/2

Samantar shreni ke vyaapak rup ka prayog karne par :

⇒ a, a+d, a+2d, a+3d, . . .

a = -1/3,

a+d = -1/3 + 1/2

= (-2+3)/6

= 1/6

a+2d = -1/3 + 2×1/2

= -1/3 + 1

= (-1+3)/3

= 2/3

a+3d = -1/3 +3×1/2

= -1/3 + 3/2

= (-2+9)/6

= 7/6

Isliye, Samantar shreni = -1/3, 1/6, 2/3, 7/6, . . . Ans.

Question-3. Yadi pratham pad (first term) a = -3 aur saarv antar d =2 ho toh Samantar Shreni ke 5 pad (five terms) gyaat kijiye.

Solution-: Diya hai: a = -3 aur d = 2

Samantar shreni ke vyaapak rup ka prayog karne par :

⇒ a, a+d, a+2d, a+3d, a+4d

a = -3, a+d = -3+2 = -1,

a+2d = -3+2×2 = 1, a+3d = -3+3×2 = 3

a+4d = -3+4×2 = 5

Isliye, Samantar shreni ke 5 pad (5 terms) = -3, -1, 1, 3, 5 Ans.

Question-4. Samantar shreni 3, 5, 7, 9, . . . . ka pratham pad (a) aur saarv antar (d) gyaat kijiye.

Solution-: Samantar shreni 3, 5, 7, 9, . . . me pratham pad (a) = 3 aur Saarv antar (d) = 5-3 = 2 Ans.

Question-5. Samantar shreni 1/2, -5/2, -11/2, -17/2, . . . . ka pratham pad (a) aur saarv antar (d) gyaat kijiye.

Solution-: Samantar shreni 1/2, -5/2, -11/2, -17/2, . . . me pratham pad (a) = 1/2 aur Saarv antar (d) = -5/2 - 1/2 = -6/2 = -3 Ans.

Question-6. Kya di gayi shreni samantar shreni hai ?

3, 3+√2, 3+2√2, 3+3√2 . . .

Solution-: Di gayi shreni 3, 3+√2, 3+2√2, 3+3√2 . . . me pratham pad (a) = 3 aur Saarv antar (d) =

a₄ - a₃ = 3+3√2 - (3+2√2)

= 3+3√2 - 3-2√2

= √2

a₃ - a₂ = 3+2√2 - (3+√2)

= 3+2√2 - 3-√2

= √2

a₂ - a₁ = 3+√2 - 3 = √2

Hum dekh sakte hai ki,

d = a₄ - a₃ = a₃ - a₂ = a₂ - a₁ = √2

Isliye, Di gayi shreni 3, 3+√2, 3+2√2, 3+3√2 Samantar shreni hai. Ans.

Question-7. Kya di gayi shreni, samantar shreni hai ?

2, 4, 8, 16 . . .

Solution-: Di gayi shreni 2, 4, 8, 16 . . . me pratham pad (a) = 2 aur Saarv antar

a₄ - a₃ = 16-8 = 8

a₃ - a₂ = 8-4 = 4

a₂ - a₁ = 4-2 = 2

d = a₄ - a₃ ≠ a₃ - a₂ ≠ a₂ - a₁

Hum dekh sakte hai ki, Saarv antar "d" ka maan equal nahi hai. Isliye di gayi shreni 2, 4, 8, 16, . . . Samantar shreni nahi hai. Ans.

Question-8. Yadi di gayi shreni, samantar shreni hai toh iske 3 terms aur gyaat kijiye:

2, 5/2, 3, 7/2, . . .

Solution-: Di gayi Samantar shreni 2, 5/2, 3, 7/2, . . . me pratham pad "a" = 2 aur

saarv antar "d" = 5/2 - 2

= (5-4)/2

= 1/2

Ab di gayi Samantar shreni ke agale 3 terms nikalne ke liye "d" ka maan last number me add karenge.

• 7/2 + 1/2 = 8/2 = 4

• 4 + 1/2 = (8+1)/2 = 9/2

• 9/2 + 1/2 = 10/2 = 5

Ab di gayi Samantar shreni ke 3 terms likhne par-:

2, 5/2, 3, 7/2, 4, 9/2, 5 . . . Ans.

Question-9. Yadi di gayi shreni, samantar shreni hai toh iske 3 terms aur gyaat kijiye:

-1.2, -3.2, -5.2, -7.2, . . .

Solution-: Di gayi Samantar shreni -1.2, -3.2, -5.2, -7.2, . . . me pratham pad "a" = -1.2 aur

saarv antar "d" = -3.2 - (-1.2)

= -3.2 + 1.2

= -2.0

Ab di gayi Samantar shreni ke agale 3 terms nikalne ke liye "d" ka maan last number me add karenge.

• -7.2 + (-2.0) = -7.2 - 2.0 = -9.2

• -9.2 + (-2.0) = -9.2 - 2.0 = -11.2

• -11.2 + (-2.0) = -11.2 - 2.0 = -13.2

Ab di gayi Samantar shreni ke 3 terms likhne par-:

-1.2, -3.2, -5.2, -7.2, -9.2, 11.2, 13.2, . . . Ans.

Question-10. Yadi ek paudhe (plant) ke lambaai 5 lagatar din (5 days) me 9.5cm, 13.5cm, 17.5cm, 21.5cm aur 25.5cm hai. Iss paudhe (plant) ki prati din (per day) lambaai me vriddhi gyaat kijiye.

Solution-: Diya hai paudhe ki 5 din (5days) me lambaai (in cm) :-

9.5, 13.5, 17.5, 21.5, 25.5

Di gayi shreni ka Saarv antar "d" gyaat karne par :-

d = 25.5 - 21.5 = 21.5 - 17.5 = 17.5 - 13.5 = 13.5 - 9.5 = 4.0

Hum dekh sakte hai ki, saarv antar "d" ka maan sabhi pado (terms) ke liye equal hai. Isliye di gayi shreni Samantar shreni me hai.

Tab paudhe me prati din lambaai me vridhi , saarv antar ke equal hogi.

Isliye, paudhe me prati din lambaai me vridhi = 4.0 cm Ans.

Ye bhi padhe-:
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