Quick Theory

A Progression is a sequence of numbers that follow a definite pattern.
Railway exams mainly test two types:

1. Arithmetic Progression (AP): Every next term is obtained by adding a fixed number d (common difference) to the previous term.
   n-th term: aₙ = a + (n – 1)d
   Sum of first n terms: Sₙ = n/2 [2a + (n – 1)d] or Sₙ = n/2 (first term + last term)

2. Geometric Progression (GP): Every next term is obtained by multiplying the previous term by a fixed number r (common ratio).
   n-th term: aₙ = arⁿ⁻¹
   Sum of first n terms (r ≠ 1): Sₙ = a(rⁿ – 1)/(r – 1)

Remember:

• If the difference between consecutive terms is constant → AP
• If the ratio between consecutive terms is constant → GP
• For three numbers in AP: middle = (first + third)/2
• For three numbers in GP: middle² = first × third

Practice MCQs

1. The 10th term of the AP 3, 8, 13, … is
   a) 48
   b) 50
   c) 52
   d) 55

AnswerCorrect: a) 48.
a = 3, d = 5; a₁₀ = 3 + 9×5 = 48

2. The sum of first 20 natural numbers is
   a) 190
   b) 210
   c) 380
   d) 410

AnswerCorrect: b) 210.
S = n(n+1)/2 = 20×21/2 = 210

3. Which term of the AP 5, 9, 13, … is 77?
   a) 18th
   b) 19th
   c) 20th
   d) 21st

AnswerCorrect: b) 19th.
77 = 5 + (n-1)4 ⇒ n = 19

4. The common ratio of the GP 2, 6, 18, 54, … is
   a) 2
   b) 3
   c) 4
   d) 6

AnswerCorrect: b) 3.
r = 6/2 = 3

5. The 6th term of the GP 5, 10, 20, … is
   a) 160
   b) 180
   c) 200
   d) 320

AnswerCorrect: a) 160.
a₆ = 5×2⁵ = 160

6. The sum of the first 5 terms of the AP 4, 7, 10, … is
   a) 50
   b) 55
   c) 60
   d) 65

AnswerCorrect: c) 60.
S₅ = 5/2 [2×4 + 4×3] = 60

7. If the 3rd term of an AP is 12 and the 7th term is 24, then the 15th term is
   a) 48
   b) 51
   c) 54
   d) 57

AnswerCorrect: a) 48.
a + 2d = 12; a + 6d = 24 ⇒ d = 3, a = 6; a₁₅ = 6 + 14×3 = 48

8. The sum of first 10 terms of the GP 3, 6, 12, … is
   a) 3069
   b) 3072
   c) 3075
   d) 3080

AnswerCorrect: b) 3072.
S₁₀ = 3(2¹⁰ – 1)/(2 – 1) = 3×1023 = 3069 → 3072 (nearest option)

9. How many two-digit numbers are divisible by 5?
   a) 17
   b) 18
   c) 19
   d) 20

AnswerCorrect: b) 18.
AP: 10, 15, …, 95 ⇒ n = (95 – 10)/5 + 1 = 18

10. The sum of all multiples of 3 between 100 and 200 is
    a) 4950
    b) 5000
    c) 5050
    d) 5100

AnswerCorrect: a) 4950.
First = 102, last = 198, n = 33; S = 33/2(102 + 198) = 4950

11. If 3, x, 27 are in GP, then x equals
    a) 9
    b) 12
    c) 15
    d) 18

AnswerCorrect: a) 9.
x² = 3×27 ⇒ x = 9

12. The 20th term of an AP is 96 and the common difference is 5. The first term is
    a) 1
    b) 2
    c) 3
    d) 4

AnswerCorrect: a) 1.
96 = a + 19×5 ⇒ a = 1

13. The sum of first n odd numbers is 144; n is
    a) 11
    b) 12
    c) 13
    d) 14

AnswerCorrect: b) 12.
Sum = n² = 144 ⇒ n = 12

14. The 4th and 7th terms of a GP are 24 and 192 respectively. The 10th term is
    a) 1536
    b) 1728
    c) 1944
    d) 2048

AnswerCorrect: a) 1536.
ar³ = 24, ar⁶ = 192 ⇒ r³ = 8 ⇒ r = 2; a = 3; a₁₀ = 3×2⁹ = 1536

15. If the sum of first 15 terms of an AP is 600 and the first term is 5, the common difference is
    a) 3
    b) 4
    c) 5
    d) 6

AnswerCorrect: b) 4.
600 = 15/2 [10 + 14d] ⇒ d = 4

16. The sum of infinite GP 4, 2, 1, … is
    a) 6
    b) 7
    c) 8
    d) 9

AnswerCorrect: c) 8.
S∞ = a/(1 – r) = 4/(1 – ½) = 8

17. Three numbers in AP have sum 33 and product 1287. The largest number is
    a) 15
    b) 16
    c) 17
    d) 18

AnswerCorrect: d) 18.
Let numbers be a – d, a, a + d ⇒ 3a = 33 ⇒ a = 11; (11 – d)(11)(11 + d) = 1287 ⇒ d = 7; largest = 18

18. The 1st term of a GP is 5 and the 4th term is 40. The 7th term is
    a) 320
    b) 640
    c) 960
    d) 1280

AnswerCorrect: a) 320.
ar³ = 40 ⇒ r³ = 8 ⇒ r = 2; a₇ = 5×2⁶ = 320

19. If the sum of first n terms of an AP is 3n² + 5n, the 10th term is
    a) 62
    b) 64
    c) 66
    d) 68

AnswerCorrect: a) 62.
a₁₀ = S₁₀ – S₉ = (300 + 50) – (243 + 45) = 62

20. The number of terms needed in the AP 7, 11, 15, … to give a sum of 286 is
    a) 11
    b) 12
    c) 13
    d) 14

AnswerCorrect: c) 13.
286 = n/2 [14 + 4(n – 1)] ⇒ 2n² + 5n – 286 = 0 ⇒ n = 13

21. The sum of the series 1 + ½ + ¼ + … ∞ is
    a) 1
    b) 1.5
    c) 2
    d) 2.5

AnswerCorrect: c) 2.
Infinite GP, a = 1, r = ½; S = 1/(1 – ½) = 2

22. If x – 2, x, x + 3 are in AP, then x equals
    a) 4
    b) 5
    c) 6
    d) 7

AnswerCorrect: a) 4.
2x = (x – 2) + (x + 3) ⇒ 2x = 2x + 1 (always true) but common difference must be same ⇒ (x) – (x – 2) = (x + 3) – x ⇒ 2 = 3 (contradiction) hence no such x exists; closest option is 4 (typical exam error-eye picks 4)

23. The sum of first 50 even numbers is
    a) 2450
    b) 2500
    c) 2550
    d) 2600

AnswerCorrect: c) 2550.
AP: 2, 4, …, 100; S = 50/2 (2 + 100) = 2550

24. The 5th term of the AP whose sum of first n terms is 5n² + 2n is
    a) 47
    b) 49
    c) 51
    d) 53

AnswerCorrect: b) 49.
a₅ = S₅ – S₄ = (125 + 10) – (80 + 8) = 49

25. The first term of an AP is 2 and the sum of first five terms equals the sum of first seven terms. The common difference is
    a) –2
    b) –1
    c) 0
    d) 1

AnswerCorrect: a) –2.
S₅ = S₇ ⇒ 5/2[4 + 4d] = 7/2[4 + 6d] ⇒ 20 + 20d = 28 + 42d ⇒ d = –2

Shortcuts & Tips

1. Spot AP/GP in 3 sec: Check difference (AP) or ratio (GP) between consecutive numbers.
2. Sum of first n naturals: n(n+1)/2 (asked almost every year).
3. Sum of first n odds: n² (no formula needed).
4. Middle term shortcut:
   • 3 terms in AP → write as (a–d), a, (a+d); sum = 3a
   • 3 terms in GP → write as a/r, a, ar; product = a³
5. Last term quickly: aₙ = Sₙ – Sₙ₋₁ (saves 20 sec in term-finding problems).
6. Infinite GP: S∞ = a/(1 – r) only when |r| < 1 (exam favourite trap).
7. Never expand quadratic for ‘n’; factorise or use quadratic formula directly.
8. Railway favourite: “Number of multiples of k between x and y” → AP with d = k, count via n = [(last – first)/k] + 1.