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Arithmetic progressions

Lesson 10 of 113

1. Ye topic hai kya?

List of numbers jahan har kadam same jod/ghatav — common difference d. Paper: 15th term kya, pehle 20 ka sum, kaun sa term 63 hai, 12 rows mein kitni seats.

Yahan lock

a, d, an = a+(n−1)d, Sn = n/2 [2a+(n−1)d], 1+2+…+n = n(n+1)/2, last term se sum, 2b=a+c, n nikalna (kabhi quadratic).

Yahan kya nahi

Pattern hunt, agla ? — Number series (wahan 4,9,14,+5). GP 5,10,20,40. Quadratic product 56. Do lines. Triangles agla geometry chapter.

2. AP kya — d lock

First term a (ya a₁). Common difference d = a₂ − a₁ = a₃ − a₂ = …. d positive, negative, ya 0 (constant list).

a=6, d=4. Terms: a, a+d, a+2d, a+3d, …

ListAP?a, d
9, 13, 17, 21haan9, +4
20, 16, 12, 8haan20, −4
11, 11, 11haan11, 0
5, 10, 20, 40nahiguna 2 — GP, d nahi
6, 10, 15, 21nahigaps +4,+5,+6
Pehle do ka farq dekh ke baaki mat assume. Teesra bhi check. 6, 10, 15 AP nahi.
Number series page pe “agla 23” type hunt. Yahan 50th term formula se, 47 terms count nahi.

3. n-th term

an = a + (n − 1)d
n=1 pe a. n=2 pe a+d. (n−1) steps, har step d.

Kyun n−1? 15th term tak 14 jumps. 15×d mat jodna.

Example · aₙ

6, 10, 14, … ka 11th term?

Solution

  1. a, d

    a=6, d=4.

  2. Formula

    a₁₁=6+(11−1)·4=6+40=46.

Answer46

Ulta: 7, 11, 15, … mein 63 kaun sa term? 7+(n−1)4=63 → (n−1)4=56 → n−1=14 → n=15.

Agar aₙ formula diya: aₙ=4n+3. Yeh AP: a₁=7, a₂=11, d=4. General: an = dn + (a−d) linear in n.

Last term ℓ = aₙ = a+(n−1)d. Sum mein kaam aayega.

4. Sum of n terms

Sₙ = a₁+a₂+…+aₙ. Do useful likhaawat:

Sn = n/2 [ 2a + (n−1)d ]
Sn = n/2 ( a + ℓ )  jab last term pata ho.
Kyun: reverse jod — har pair a+ℓ, n/2 pairs.
Special: 1+2+…+n = n(n+1)/2 — ye AP hai a=1, d=1. Same formula: n/2[2+(n−1)] = n(n+1)/2.
Example · Sₙ

6, 10, 14, … ke pehle 15 terms ka sum?

Solution

  1. Plug

    n=15, a=6, d=4. S=15/2 [12 + 14·4]=15/2 [12+56]=15/2 · 68.

  2. Guna

    15×34=510. Check last: a₁₅=6+14·4=62. S=15/2(6+62)=15/2·68=510.

Answer510
an = Sn − Sn−1 (n≥2). S₁=a.
Diya Sₙ=n²+3n to aₙ=(n²+3n)−((n−1)²+3(n−1)). Exam trick.

Sₙ=n²+3n: aₙ=n²+3n − (n²−2n+1+3n−3)=n²+3n−(n²+n−2)=2n+2. d=2, AP.

S₁₀ last term nahi — sum hai. a₁₀ alag. n/2 bahar, [ ] ke andar 2a+(n−1)d. n/2 × 2a = na, phir n/2(n−1)d.

5. n kitna — term se ya sum se

aₙ diya: linear, n nikalna seedha. Sₙ diya: n ke 2 power — quadratic. D≥0, n positive integer. Extra root usually negative discard.

Example · Sₙ se n

3, 7, 11, … ka sum 210. Kitne terms?

Solution

  1. Sₙ=210

    a=3, d=4. n/2 [6+(n−1)4]=210 → n/2(6+4n−4)=210 → n/2(2+4n)=210.

  2. Simplify

    n(1+2n)=210 → 2n²+n−210=0. D=1+1680=1681=41². n=(−1+41)/4=10. (2a=4.)

  3. Check

    S₁₀=10/2[6+36]=5×42=210. Negative root chhodo.

Answer10 terms
Quadratic equations page ka formula yahan n ke liye. n fraction aaye to wo Sₙ is AP ka itna terms nahi — sawal/data check.

6. Teen terms — 2b = a+c

a, b, c AP ⇔ 2b = a + c (common d: b−a=c−b).
Teen numbers AP, sum 3A: likho A−d, A, A+d. Product/sum se d.

5, x, 17 AP: 2x=5+17=22, x=11. d=6.

AMs beech mein: 4 aur 28 ke beech 5 arithmetic means = total 7 terms. d=(28−4)/6=4. Means: 8, 12, 16, 20, 24.

Odd number of terms: middle = average of first aur last = Sₙ/n.

7. Word — seats, saving

Pehli quantity a, har next mein same +d. Rows, months, logs. Consecutive integers product Quadratic. Do prices Pair of linear.

Example · rows

Sabha: pehli row 8 seats, har agle row mein 2 extra. 12 rows. Kul seats?

Solution

  1. AP

    a=8, d=2, n=12.

  2. S

    12/2 [16+11·2]=6[16+22]=6×38=228.

Answer228
Example · saving

Pehle hafte ₹200, har agle hafte ₹25 zyada. 8 hafte. Kul?

Solution

  1. a=200, d=25, n=8

    S=8/2 [400+7·25]=4[400+175]=4×575=2300.

Answer₹2300

8. Sawal — basic se pro

Q1 · Basic

Kaun si AP hai?

  1. 6, 10, 15, 21
  2. 9, 13, 17, 21
  3. 5, 10, 20, 40
  4. 2, 3, 5, 8

Solution

  1. d same?

    B: +4,+4,+4. A: +4,+5,+6. C: GP ×2. D: Fibonacci gaps.

AnswerB
Q2 · Basic

6, 10, 14, … ka 11th term?

  1. 46
  2. 50
  3. 44
  4. 40

Solution

  1. 6+(11−1)×4

    10 jumps, 6+40=46. B: 6+11×4=50 (n−1 bhool). D: sirf 10d.

AnswerA · 46
Q3 · Basic

20, 16, 12, … ka 8th term?

  1. −8
  2. −4
  3. 0
  4. 4

Solution

  1. a=20, d=−4

    20+7(−4)=20−28=−8.

AnswerA · −8
Q4 · Basic · 2b

5, x, 17 AP. x?

  1. 11
  2. 12
  3. 10
  4. 22

Solution

  1. 2x=5+17

    x=11. D sum bina aadha kiye.

AnswerA · 11
Q5 · Medium · Sₙ

6, 10, 14, … pehle 15 terms ka sum?

  1. 510
  2. 480
  3. 62
  4. 930

Solution

  1. 15/2 × 68

    510. C last term a₁₅. D: n(n+1) style galat.

AnswerA · 510
Q6 · Medium

7, 11, 15, … mein 63 kaun sa term?

  1. 14
  2. 15
  3. 16
  4. 13

Solution

  1. 7+(n−1)4=63

    n−1=14, n=15. A: (n−1) ko n samajh liya.

AnswerB · 15th
Q7 · Medium

aₙ = 4n + 3. d?

  1. 4
  2. 3
  3. 7
  4. 1

Solution

  1. a₁=7, a₂=11

    d=4. Coeff of n hi d. Constant 3 first term nahi (a₁=7).

AnswerA · 4
Q8 · Medium · seats

Pehli row 8, har agle +2, 12 rows. Kul seats?

  1. 228
  2. 240
  3. 192
  4. 30

Solution

  1. S₁₂

    6×38=228. D last row 8+22=30, sum nahi.

AnswerA · 228
Q9 · Pro · n from S

3, 7, 11, … ka sum 210. n?

  1. 10
  2. 12
  3. 15
  4. 8

Solution

  1. 2n²+n−210=0

    n=10. S₈=8/2[6+28]=4×34=136≠210. S₁₂=12/2[6+44]=6×50=300.

AnswerA · 10
Q10 · Pro

Sₙ = n² + 3n. a₅?

  1. 40
  2. 12
  3. 8
  4. 28

Solution

  1. a₅=S₅−S₄

    S₅=25+15=40, S₄=16+12=28, a₅=12. A S₅ hai. C a₁=S₁=4, d=2 → a₅=4+8=12.

AnswerB · 12

Practice

Pehle khud, phir neeche kholo.

12345 678910
P1

11, 11, 11, 11. d?

  1. 11
  2. 0
  3. 1
  4. AP nahi

Solution

  1. Farq 0

    Constant AP, d=0. aₙ=11.

AnswerB · 0
P2

a=9, d=4. a₇?

  1. 33
  2. 37
  3. 28
  4. 13

Solution

  1. 9+6×4

    33. B: 7d jod diya 9+28=37.

AnswerA · 33
P3

ℓ=62, a=6, n=15. Sₙ? (6,10,14… wahi)

  1. 510
  2. 68
  3. 930
  4. 62

Solution

  1. n/2(a+ℓ)

    15/2(6+62)=510. B a+ℓ without n/2.

AnswerA
P4

4 aur 28 ke beech 5 AMs. Pehla mean?

  1. 8
  2. 6
  3. 10
  4. 4

Solution

  1. 7 terms, d=(28−4)/6=4

    4, 8, 12, … pehla mean 8.

AnswerA · 8
P5

Pehle hafte 200, +25, 8 hafte. Kul ₹?

  1. 2300
  2. 1600
  3. 375
  4. 1800

Solution

  1. 4×575

    2300. C last week 200+175=375.

AnswerA · 2300
P6

Kaun sa aₙ AP nahi (n=1,2,3…)?

  1. 3n+1
  2. n²
  3. 5−2n
  4. 7

Solution

  1. n²: 1,4,9,16

    gaps +3,+5,+7. Linear in n = AP. Constant 7 = d=0.

AnswerB · n²
P7

Sₙ=n²+3n. S₃?

  1. 18
  2. 12
  3. 6
  4. 9

Solution

  1. 9+9=18

    a₁+a₂+a₃. aₙ=2n+2: 4+6+8=18.

AnswerA · 18
P8

Odd terms, n=9, a=3, ℓ=35. Middle term?

  1. 19
  2. 35
  3. 16
  4. 9

Solution

  1. (a+ℓ)/2

    (3+35)/2=19. 5th term. d=(35−3)/8=4, a₅=3+16=19.

AnswerA · 19
P9

20, 16, 12, … pehle 8 ka sum?

  1. 48
  2. 32
  3. −8
  4. 80

Solution

  1. 8/2[40+7(−4)]

    4[40−28]=4×12=48. C a₈. D na×a.

AnswerA · 48
P10

12 rows, pehli 8, +2. Last row seats?

  1. 30
  2. 28
  3. 32
  4. 228

Solution

  1. a₁₂=8+11×2

    30. D total sum tha. B: 10d.

AnswerA · 30

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Agle topic: Triangles.