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Number Series

Lesson 61 of 113

1. Ye topic hai kya?

Number series = numbers ki ek line, ek chhupa rule. Tumhe rule dhundh ke agla / beech ka / galat number nikalna hai.

Maths ke AP–GP chapter se farq: yahan formula ratna nahi, pattern hunt hai. 20 second mein difference, nahi to multiply, nahi to squares.

Teen sawaal types

Next: last ke baad kya?
Missing: beech ka ?
Wrong: kaun sa term pattern todta hai

Yahan kya nahi

Letter Series alag. Figure-grid ? — Missing number. AP ka Sₙ Maths pe. Number-letter mix: number yahan, letter wahan.

2. Paper pe dimag kaise chale (40 second)

Yahi order — skip mat karo:
1) Consecutive terms ka difference (+ ya −)
2) Difference ka difference (second gap constant?)
3) Ratio — ×2, ×3, ÷2…
4) ×n ± k — 5, 6, 14, 45 wala family
5) Term square/cube/prime ke kareeb?
6) Odd positions alag, even alag (do series)
7) Tab digit / reverse / factorial

Har step pe saari terms check. Pehle 2 terms se rule ghad ke baaki pe fail ho to rule galat — naya try.

Galati: 3 terms dekh ke next likh dena. LDC mein 5–6 terms hoti hain taaki do rules mein se ek mar jaye. Poori line fit karni hai.

3. Constant gap (pattern, Sₙ nahi)

Har kadam same jod/ghatav. Boxes ke beech gap likho. Gap same = constant. Agla = last + wahi gap. Maths wala Sₙ = n/2[2a+(n−1)d] yahan nahi — sirf agla / galat term.

3
+5
8
+5
13
+5
18
+5
?

Agla 18+5 = 23. Minus series bhi same: 40, 34, 28, 22 → −6 → 16.

Basic · constant gap

Q1. 4, 9, 14, 19, 24, ? Agla number kya hai?

Solution shuru

  1. Step 1 — gaps
    4
    +5
    9
    +5
    14
    +5
    19
    +5
    24
  2. Step 2 — rule

    Har gap +5. AP. Multiply ki zaroorat nahi.

  3. Step 3 — next

    24 + 5 = 29.

Final answer — yahan khatam29

Second difference constant

Pehle gaps alag hain, lekin un gaps ke gaps same. Yeh hidden quadratic hai — LDC mein “+2, +4, +6, +8” bahut aata hai.

Basic · 2nd gap

Q2. 2, 6, 12, 20, 30, ?

Solution shuru

  1. Step 1 — pehli gaps

    6−2=4, 12−6=6, 20−12=8, 30−20=10. Gaps: 4, 6, 8, 10 — constant nahi.

  2. Step 2 — doosri gaps

    6−4=2, 8−6=2, 10−8=2. Second gap hamesha +2. Agli pehli-gap = 10+2 = 12.

  3. Step 3 — next term

    30 + 12 = 42.

  4. Step 4 — doosri nazar (optional)

    2=1×2, 6=2×3, 12=3×4, 20=4×5, 30=5×6, agla 6×7=42. Same answer. Jo jaldi dikhe woh use.

Final answer — yahan khatam42

Seekh: pehli gap fail ho to turant second gap. Time bachta hai.

4. Multiply / divide (GP)

Numbers jaldi bade (2, 4, 8, 16…) ya jaldi chhote (81, 27, 9, 3) to pehle × aur ÷ socho, + nahi.

Check: har term ÷ previous. Same ratio = GP.

Basic · GP

Q3. 3, 9, 27, 81, ?

Solution shuru

  1. Step 1 — + se try

    +6, +18, +54 — yeh + nahi, har baar ×3 jaisa grow.

  2. Step 2 — ratio

    9÷3=3, 27÷9=3, 81÷27=3. Har kadam ×3.

  3. Step 3 — next

    81 × 3 = 243.

Final answer — yahan khatam243

Divide wala: 64, 32, 16, 8, ? → ÷2 → 4.

5. ×n ± k — LDC ka favourite

Na shuddha AP, na shuddha GP. Har step: pehle guna, phir chhota jod/ghatav.

  • ×1+1, ×2+2, ×3+3, ×4+4…
  • ×2+1, ×2+2, ×2+3, ×2+4…
  • ×2−1, ×2+1, ×2−1, ×2+1 (plus-minus palat)

Kaise pakdo: consecutive ka ratio ~2 ya 3, lekin exact nahi. Remainder next − (prev × n) chhota integer hota hai.

Medium · ×n+n

Q4. 5, 6, 14, 45, 184, ?

Solution shuru

  1. Step 1 — AP/GP?

    Gaps +1, +8, +31, +139 — na constant, na × same. GP nahi (6÷5=1.2, 14÷6≈2.3).

  2. Step 2 — × phir +

    5×1 + 1 = 6. Fit.

    6×2 + 2 = 14. Fit.

    14×3 + 3 = 45. Fit.

    45×4 + 4 = 184. Fit. Poori line.

  3. Step 3 — next

    184 × 5 + 5 = 920 + 5 = 925.

Final answer — yahan khatam925

Pattern: ×1+1, ×2+2, ×3+3… Index 1 se chalu. Pehli 2 terms se andaza, teesri pe lock.

6. Square, cube, prime, factorial

Exam hall mein yeh list dimag mein honi chahiye — calculate mat karo har baar.

YaadValues
Squares 1–121, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
Cubes 1–101, 8, 27, 64, 125, 216, 343, 512, 729, 1000
Primes2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37
n!1, 2, 6, 24, 120, 720, 5040

Variants: n²+1, n²−1, n³+n, prime+1, consecutive primes.

Basic · n²+1

Q5. 2, 5, 10, 17, 26, ?

Solution shuru

  1. Step 1 — gaps

    +3, +5, +7, +9 — odd numbers. Agli gap +11.

  2. Step 2 — square nazar

    1+1=2, 4+1=5, 9+1=10, 16+1=17, 25+1=26. Har term = n²+1 (n=1,2,3,4,5).

  3. Step 3

    n=6: 36+1=37. Gap se: 26+11=37. Dono same.

Final answer — yahan khatam37

Cube example: 1, 8, 27, 64, 125, ? → 6³ = 216.

Factorial: 1, 2, 6, 24, 120, ? → 6! = 720.

Prime: 2, 3, 5, 7, 11, 13, ? → 17.

7. Alternate — do series ek line mein

Odd seats ek rule, even seats doosra. Tab dikhta hai jab consecutive gaps “paagal” hon — +3, −1, +6, −2…

Alag karo: positions 1,3,5,7 aur 2,4,6,8.

Medium · Alternate

Q6. 2, 5, 4, 10, 8, 20, 16, ?

Solution shuru

  1. Step 1 — consecutive

    +3, −1, +6, −2, +12, −4 — ek rule nahi. Alternate try.

  2. Step 2 — odd positions

    2, 4, 8, 16 → ×2. Agli odd term 32, lekin ? position 8 = even.

  3. Step 3 — even positions

    5, 10, 20, ? → ×2. Agli even = 40.

  4. Step 4 — check

    Line: 2, 5, 4, 10, 8, 20, 16, 40. Odd ×2, even ×2. Fit.

Final answer — yahan khatam40

Pehle count karo ? odd hai ya even. Galat series mat nikaalna.

8. Digit / reverse (kabhi-kabhi)

Tab socho jab ×n±k fail ho aur numbers “weird” hon: 13, 31, 24, 42…

  • Reverse: 13 → 31, 24 → 42
  • Digit sum: 16 → 1+6=7, phir 16+7=23
  • n² ke digits jodna — rare at LDC, pro papers mein
Pehle classic patterns khatam karo. Digit last option — overfit easy hai.

9. Wrong number

Sawaal: series mein kaun sa galat hai (options mein terms). Next mat nikaalo.

Rule woh jo ek term hata/badal ke baaki saari terms fit kare. Jo term fit nahi, wahi wrong.
Medium · Wrong

Q7. 2, 3, 7, 22, 89, 445, 2677 — kaun sa number galat hai?

Solution shuru

  1. Step 1 — ×n+1 try

    2×1 + 1 = 3. Theek.

    3×2 + 1 = 7. Theek.

    7×3 + 1 = 22. Theek.

    22×4 + 1 = 89. Theek.

  2. Step 2 — tod

    89×5 + 1 = 445+1 = 446. Line mein 445 likha hai. Yahan toot.

  3. Step 3 — aage check

    Agar 446 hota: 446×6 + 1 = 2676+1 = 2677. Last term theek. Sirf 445 galat.

Final answer — yahan khatam445 (hona chahiye tha 446)

Wrong-number mein last term bhi galat ho sakta hai — poori chain chalao.

10. Beech ka missing

? middle mein. Dono taraf se rule lock karo — left se next, right se previous, match hona chahiye.

Medium · Missing

Q8. 4, 9, 19, ?, 79, 159

Solution shuru

  1. Step 1 — left se

    4×2+1=9, 9×2+1=19. Rule: ×2+1. To ? = 19×2+1 = 39.

  2. Step 2 — right se check

    39×2+1=79. 79×2+1=159. Dono taraf fit. Case zinda.

  3. Step 3 — AP to nahi?

    Gaps +5, +10, … +? Doubling gaps would give +20 then +40: 19+20=39, 39+40=79, 79+80=159. Yeh bhi same 39. Extra confusion nahi.

Final answer — yahan khatam39

11. Digit / reverse (line)

Kabhi term ke digits ka sum, reverse, ya × apne digits. Figure-grid (3×3 box) yahan nahi — Missing number.

13, 31, 12, 21, 14, ? → har pair reverse: 13↔31, 12↔21, 14↔41.

Medium · reverse pair

Q9. 13, 31, 12, 21, 14, ?

Solution shuru

  1. Step 1 — consecutive

    +18, −19, +9, −7, +? — koi constant gap nahi. × bhi nahi.

  2. Step 2 — jodi

    (13,31) reverse. (12,21) reverse. Agli jodi 14 ka reverse 41.

  3. Step 3 — grid nahi

    3×3 box ka row-sum Missing number pe. Yahan line.

Final answer — yahan khatam41

12. Pro checklist (exam hall)

  1. 5+ terms hain? Pehle 2 se rule, baaki pe verify. Fail = naya rule.
  2. Numbers slow badhein to +, jaldi badhein to ×.
  3. Second gap constant dikhe to quadratic / n(n+1) socho.
  4. Consecutive paagal, alternate shaant — do series.
  5. Wrong-number: next mat likho; ek term replace karke chain chalao.
  6. Do rules dono fit karein to jo saari terms pe chale, aur options mein ho, wahi. Rare tie = simpler rule (AP before cubic).
  7. Calculator nahi — 184×5+5 jaise pro step ko tod: 180×5=900, 4×5=20, +5=925.
Teen terms ki GP aur AP dono possible hon (2, 4, 8 agla 16 ya 14?) — chauthi term dekh. Agar nahi di, options dekho kaun sa series ke baaki feel se match.

13. Practice — basic se pro, poora solution

Pehle khud socho, phir neeche solution kholo. Upar Q1–Q9 theory ke saath.

Basic · constant gap

Practice 1. 11, 17, 23, 29, 35, ?

Solution shuru

  1. Step 1 — gaps

    17−11=6, 23−17=6, 29−23=6, 35−29=6. Sab +6.

  2. Step 2 — next

    35+6=41.

Final answer — yahan khatam41
Basic · add previous

Practice 2. 3, 5, 8, 13, 21, ?

Solution shuru

  1. Step 1 — constant gap?

    +2, +3, +5, +8 — gap khud badh raha, Q5 wala n²+1 (2,5,10,17) nahi.

  2. Step 2 — pichhle do

    3+5=8, 5+8=13, 8+13=21. Har term = pichhle do ka sum.

  3. Step 3 — next

    13+21=34.

Final answer — yahan khatam34
Basic · GP

Practice 3. 4, 8, 16, 32, 64, ?

Solution shuru

  1. Step 1 — ratio

    8÷4=2, 16÷8=2, 32÷16=2, 64÷32=2. ×2.

  2. Step 2

    64×2=128.

Final answer — yahan khatam128
Medium · ×n+n

Practice 4. 3, 4, 10, 33, 136, ?

Solution shuru

  1. Step 1 — AP/GP nahi

    +1, +6, +23, +103. Ratio 4/3, 10/4=2.5, 33/10=3.3 — GP nahi.

  2. Step 2 — ×n+n

    3×1+1=4. 4×2+2=10. 10×3+3=33. 33×4+4=136. Poori line lock.

  3. Step 3 — next

    136×5+5. 130×5=650, 6×5=30, +5 → 685.

Final answer — yahan khatam685
Medium · + 3ⁿ

Practice 5. 8, 9, 12, 21, 48, ?

Solution shuru

  1. Step 1 — gaps

    +1, +3, +9, +27. Yeh 3 ki powers: 3⁰, 3¹, 3², 3³.

  2. Step 2 — next gap

    3⁴ = 81. 48+81=129.

  3. Step 3 — × se to nahi?

    8×1+1=9, 9×1+3=12, 12×1+9=21, 21×2+6=48? 21+27=48, ×2+6 extra ghadna. Gaps wala rule saaf hai. Wahi.

Final answer — yahan khatam129
Medium · Alternate

Practice 6. 3, 10, 4, 16, 5, 22, 6, ?

Solution shuru

  1. Step 1 — consecutive fail

    +7, −6, +12, −11, +17, −16 — do palat series.

  2. Step 2 — odd seats

    3, 4, 5, 6 → +1. Agli odd 7, lekin ? even seat (8th).

  3. Step 3 — even seats

    10, 16, 22, ? → +6. 22+6=28.

Final answer — yahan khatam28
Medium · ×2+k

Practice 7. 5, 11, 24, 51, 106, ?

Solution shuru

  1. Step 1 — almost ×2

    5×2=10, line 11 → +1 extra. 11×2=22, line 24 → +2. Pattern: ×2+1, ×2+2, ×2+3…

  2. Step 2 — verify

    5×2+1=11. 11×2+2=24. 24×2+3=51. 51×2+4=106. Fit.

  3. Step 3 — next

    106×2+5=212+5=217.

Final answer — yahan khatam217
Pro · Wrong

Practice 8. 1, 4, 9, 16, 26, 36, 49 — galat number kaun sa?

Solution shuru

  1. Step 1 — squares?

    1=1², 4=2², 9=3², 16=4², 36=6², 49=7². Beech 26 — 5²=25 hona chahiye.

  2. Step 2 — gaps se

    +3, +5, +7, +10, +10, +13. Square series mein +odd: +3,+5,+7,+9,+11,+13. +10/+10 toot — 16 ke baad +9=25, phir +11=36.

  3. Step 3

    Sirf 26 replace: 25. Baaki saari squares theek.

Final answer — yahan khatam26 (hona chahiye 25)
Basic · Missing

Practice 9. 7, 14, 28, ?, 112, 224

Solution shuru

  1. Step 1 — left

    7×2=14, 14×2=28. ×2. ? = 28×2=56.

  2. Step 2 — right

    56×2=112, 112×2=224. Fit.

Final answer — yahan khatam56
Pro · ×n+n

Practice 10. 2, 3, 8, 27, 112, ?

Solution shuru

  1. Step 1 — na AP na GP

    +1, +5, +19, +85. Ratio ~1.5, 2.6, 3.4, 4.1 — n badh raha.

  2. Step 2 — ×n+n

    2×1+1=3. 3×2+2=8. 8×3+3=27. 27×4+4=112. Lock.

  3. Step 3 — next

    112×5+5. 100×5=500, 12×5=60, +5 → 565.

Final answer — yahan khatam565

Yahi family Q4 / Practice 4 ki hai. LDC pro set mein yeh pattern baar-baar.

14. Itna kaafi hai kya?

Agar tum:

  • pehle difference, phir ratio, phir ×n±k order se try kar sako
  • second gap aur alternate series alag kar sako
  • wrong vs missing vs next teen types ghalat na milao
  • squares 1–12, cubes 1–10, primes 30 tak yaad hon

to JA/CA-II ke number series — basic se pro tak — attempt ho sakte hain.

Agle topic: Letter Series.

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