Samantar Shredhiyan (Arithmetic Progression) Notes Class 10 — Simple Hindi Me

arithmetic progression class 10 notes dhoondh rahe ho? To sahi jagah aaye ho. Class 10 Maths ka ye chapter (NCERT me “Samantar Shredhiyan” ke naam se) board exam me pakke 5-8 marks ka hota hai — aur achhi khabar ye hai ki ye chapter samajhne me sabse aasaan hai. Bas 2 formula yaad karne hain, aur poora chapter unhi par chalta hai.

Dekho, kai students AP ko ratne ki koshish karte hain — formula to yaad ho jaata hai, lekin question thoda ghumakar aaye to haath ruk jaata hai. Is notes me hum concept ko aise samjhenge ki koi bhi question aaye, tumhe lage “arey, ye to wahi hai.”

Arithmetic Progression Class 10 Notes — Basic Concept Samjho

Arithmetic Progression (AP) numbers ki aisi list hoti hai jisme har term apne pehle wale term se ek FIXED number jodkar (ya ghatakar) banta hai. Is fixed number ko common difference kehte hain, jise “d” se likhte hain.

Example dekho: 2, 5, 8, 11, 14, …
Yahan 2 me 3 jodne par 5 milta hai, 5 me 3 jodne par 8, aur aise hi aage. To ye ek AP hai jisme:

• Pehla term (a) = 2
• Common difference (d) = 3
  Ek aur example: 10, 7, 4, 1, -2, … — yahan har baar 3 GHAT raha hai, to d = -3. Haan, common difference negative bhi ho sakta hai, aur zero bhi (jaise 5, 5, 5, 5 — yahan d = 0).
  AP ka general form hota hai: a, a+d, a+2d, a+3d, …

AP Pehchano — Quick Test

Neeche wali lists me se kaun AP hai?

1. 1, 3, 5, 7, 9 — Haan (d = 2)
2. 2, 4, 8, 16 — Nahi (difference badal raha hai: 2, 4, 8)
3. 5, 5, 5, 5 — Haan (d = 0)
4. 100, 90, 80, 70 — Haan (d = -10)
   Yaad rakho: sirf tab AP hai jab LAGATAR terms ka difference SAME ho.

AP Ke 2 Sabse Important Formula

1. nth Term Ka Formula (kisi bhi term ko nikalna)

a(n) = a + (n-1)d
Yahan a = pehla term, d = common difference, n = kaun sa term chahiye.
Example: AP 2, 7, 12, 17, … ka 10th term nikalo.

• a = 2, d = 5, n = 10
• a(10) = 2 + (10-1) x 5 = 2 + 45 = 47

2. Pehle n Terms Ke Sum Ka Formula

S(n) = n/2 x [2a + (n-1)d]
Iska ek aur roop bhi hai, jab aakhri term (l) pata ho:
S(n) = n/2 x (a + l)
Example: AP 2, 7, 12, … ke pehle 15 terms ka sum nikalo.

• a = 2, d = 5, n = 15
• S(15) = 15/2 x [2×2 + (15-1)x5] = 15/2 x [4 + 70] = 15/2 x 74 = 15 x 37 = 555

Ek Aur Kaam Ka Formula

Agar a, b, c AP me hain, to: 2b = a + c
Matlab beech wala term dono taraf ke terms ka average hota hai. Ise “arithmetic mean” kehte hain. Examp

le: 4, 9, 14 me 2×9 = 18 aur 4+14 = 18. Barabar!

Solved Examples — NCERT Pattern Par

Example 1: AP 3, 8, 13, 18, … ka kaun sa term 78 hai?

• a = 3, d = 5. Maan lo nth term 78 hai.
• 3 + (n-1) x 5 = 78
• (n-1) x 5 = 75
• n-1 = 15, to n = 16
• Jawab: 16th term 78 hai.

Example 2: AP 8, 3, -2, … ke pehle 22 terms ka sum nikalo.

• a = 8, d = 3 – 8 = -5, n = 22
• S(22) = 22/2 x [2×8 + (22-1)x(-5)]
• = 11 x [16 – 105] = 11 x (-89) = -979
• Jawab: -979 (haan, sum negative bhi aa sakta hai — ghabrao mat!)

Example 3: Kisi AP ka 7th term 34 hai aur 13th term 64 hai. AP batao.

• a + 6d = 34 … (i)
• a + 12d = 64 … (ii)
• (ii) – (i): 6d = 30, to d = 5
• (i) me rakho: a + 30 = 34, to a = 4
• Jawab: AP hai 4, 9, 14, 19, …

Example 4: Kya 0, AP 40, 37, 34, … ka koi term hai?

• a = 40, d = -3. Maan lo nth term 0 hai.
• 40 + (n-1) x (-3) = 0
• (n-1) x (-3) = -40
• n-1 = 40/3 = 13.33…
• n poora number (natural number) nahi aaya, isliye 0 is AP ka term NAHI hai.

Ye chaaro examples NCERT exercise ke pattern par hain — board me bilkul aise hi sawaal aate hain.

NCERT Exam Tips — AP Me Full Marks Kaise Laayein

1. d nikalte waqt sign ka dhyan rakho: 8, 3, -2 me d = 3 – 8 = -5 hai, 5 nahi. Sign ki galti sabse common hai.
2. “Kaun sa term” wale sawaal me n hamesha natural number aana chahiye. Agar fraction/decimal aaye to jawab hai “ye term nahi hai” (Example 4 dekho).
3. Sum ke formula ke dono roop yaad rakho. Jab last term diya ho to S(n) = n/2 (a + l) zyada aasaan padta hai.
4. Word problems me pehle a aur d nikalo: “Pehle 3 terms ka sum 27 hai” jaise sawaalon me a + (a+d) + (a+2d) = 27 likhkar solve karo — seedha equation banao.
5. Answer check karo: nth term nikalne ke baad ek-do term aage-peechhe likhkar dekho ki pattern match ho raha hai ya nahi.

Quick Revision — Ek Nazar Me

• AP: har consecutive term ka difference same (d)
• General form: a, a+d, a+2d, …
• nth term: a(n) = a + (n-1)d
• Sum of n terms: S(n) = n/2 [2a + (n-1)d] = n/2 (a + l)
• a, b, c AP me hon to 2b = a + c
• d positive, negative ya zero — teeno possible
• n hamesha 1, 2, 3, … (natural number) hota hai

FAQs

Q1. AP aur GP me kya farq hai?
AP me har term me FIXED NUMBER judta/ghatta hai (jaise 2, 5, 8), jabki GP (Geometric Progression, Class 11 me padhoge) me har term FIXED NUMBER se multiply hota hai (jaise 2, 6, 18). Class 10 me sirf AP hai.

Q2. Kya d zero ho sakta hai?
Haan. 7, 7, 7, 7 bhi ek AP hai jisme d = 0. Board me aisa sawaal kabhi-kabhi concept check karne ke liye aata hai.

Q3. AP ke questions me sabse zyada marks kahan kat-te hain?
Sign ki galti (d = -5 ko 5 likh dena), n ko fraction me chhod dena, aur sum ke formula me 2a ki jagah a likh dena — ye teen galtiyan sabse zyada hoti hain.

Q4. Kya AP ka chapter trigonometry se juda hai?
Nahi, AP algebra ka chapter hai. Lekin Class 10 ke aur chapters ke notes bhi yahan milenge.

Maths ke aur notes: Quadratic Equations Class 10 Notes | Trigonometry Formulas Class 10