20. Arithmetic and Geometric Sequences

CSS GSA — Paper 7, Part-II, Section VI, Sub-topic 20 (Arithmetic and Geometric Sequences)

Past Paper Weightage

• Moderate in MPT/General Ability papers — finding a specific term or the sum of a sequence is a standard, formulaic question type.

Arithmetic Progression (AP)

• A sequence where each term increases/decreases by a constant difference (d) from the previous term (e.g., 2, 5, 8, 11, ... where d=3)
• nth term formula:

$$a_n = a_1 + (n-1)d$$

where a₁ = first term, d = common difference, n = term position

• Sum of first n terms:

$$S_n = \frac{n}{2}\left[2a_1 + (n-1)d\right] = \frac{n}{2}(a_1+a_n)$$

(average of first and last term, times the number of terms)

Worked Example 1 (AP — Find a Term)

• Find the 15th term of the AP: 4, 9, 14, 19, ...
  • Step 1: Identify a₁ = 4 and the common difference d = 9−4 = 5.
  • Step 2: Apply the formula: a₁₅ = a₁ + (15−1)d = 4 + 14×5.
  • Step 3: Calculate: 14×5 = 70, so a₁₅ = 4 + 70 = 74.

Worked Example 2 (AP — Sum of Terms)

• Find the sum of the first 20 terms of the AP: 3, 7, 11, 15, ...
  • Step 1: Identify a₁ = 3, d = 7−3 = 4, n = 20.
  • Step 2: Apply the sum formula: S₂₀ = (20/2) × [2(3) + (20−1)(4)] = 10 × [6 + 76].
  • Step 3: Calculate: 6+76=82, so S₂₀ = 10 × 82 = 820.

Geometric Progression (GP)

• A sequence where each term is found by multiplying the previous term by a constant ratio (r) (e.g., 2, 6, 18, 54, ... where r=3)
• nth term formula:

$$a_n = a_1 \times r^{(n-1)}$$

• Sum of first n terms (r ≠ 1):

$$S_n = a_1 \times \frac{r^n - 1}{r - 1}$$

Worked Example 3 (GP — Find a Term)

• Find the 6th term of the GP: 3, 6, 12, 24, ...
  • Step 1: Identify a₁ = 3 and the common ratio r = 6÷3 = 2.
  • Step 2: Apply the formula: a₆ = a₁ × r^(6−1) = 3 × 2⁵.
  • Step 3: Calculate: 2⁵ = 32, so a₆ = 3×32 = 96.

Worked Example 4 (GP — Sum of Terms)

• Find the sum of the first 5 terms of the GP: 2, 6, 18, 54, ...
  • Step 1: Identify a₁ = 2, r = 6÷2 = 3, n = 5.
  • Step 2: Apply the sum formula: S₅ = a₁ × (r⁵−1)/(r−1) = 2 × (3⁵−1)/(3−1).
  • Step 3: Calculate 3⁵ = 243, so S₅ = 2 × (243−1)/2 = 2 × 242/2 = 2 × 121 = 242.

MCQ Traps

• AP has a constant DIFFERENCE (addition/subtraction between terms); GP has a constant RATIO (multiplication/division between terms) — check which pattern applies before picking a formula
• In the AP nth-term formula, the exponent-like term is (n−1), NOT n — a very common off-by-one error
• In the GP formulas, the same (n−1) offset applies to the exponent for finding a term, but the FULL n is used in the sum formula's exponent (rⁿ)

Self-Test Questions

1. Find the 12th term of the AP: 7, 12, 17, 22, ... showing the formula and calculation.
2. Find the sum of the first 10 terms of the AP: 5, 8, 11, 14, ... showing each step.
3. Find the 5th term of the GP: 4, 8, 16, 32, ... showing the formula and calculation.
4. Find the sum of the first 4 terms of the GP: 5, 15, 45, 135, ... showing each step.

Past-Paper MCQs (FPSC)

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The FPSC MCQ papers you shared (2024 and 2025) had no direct sequence question.

Past-Paper Questions with Answers (FPSC)

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Every question below appeared in the FPSC papers you shared; the year and question number are in each title. Answers are worked out by me. Number-series questions are asked almost every year.

• CSS 2024 Q6(c): Find the missing numbers. (5)

  Series	Rule	Answer
  121, 11, 81, 9, ?, 7	Square, then its root	49
  100, 50, 25, ?, 6.25	Halve each time	12.5
  4, 9, 64, 125, 1296, ?	2², 3², 4³, 5³, 6⁴, 7⁴	2401
  2, 5, 12, 24, 48, ?	Doubles from 12 onward	96
  44, 22, 66, 33, 132, ?	÷2, ×3, ÷2, ×4, ÷2	66

• CSS 2023 Q6(b): Find the missing number in each series. (5)

  Series	Rule	Answer
  1, 8, 4, 27, 9, ?	1³, 2³, 2², 3³, 3², 4³	64
  3, 6, 8, 16, 18, ?	×2, +2, ×2, +2, ×2	36
  2, 8, 512, ?	Cube each time (2³ = 8, 8³ = 512)	134,217,728 (512³)
  81, 9, 64, 8, ?, 12	Square, then its root	144
  6, 11, 21, 36, 56, ?	Differences 5, 10, 15, 20, 25	81

• CSS 2022 Q7(b): Find the missing number. (5)

  Series	Rule	Answer
  4, 16, 36, 64, ?, 144	Squares of even numbers	100
  30, 29, 27, ?, 20, 15	Subtract 1, 2, 3, 4, 5	24
  1, 7, 15, 25, ?, 51	Differences 6, 8, 10, 12, 14	37
  0, 2, 6, 12, 20, 30, ?	n(n + 1); differences 2, 4, 6…	42
  48, 24, 72, 36, 108, ?	÷2, ×3 alternately	54

• CSS 2021 Q8(d): Find the missing numbers in the series. (5)

  Series	Rule	Answer
  4, 18, ?, 100, 180, 294	First differences 14, 30, 52, 80, 114 (second differences up by 6)	48
  15, 31, 63, 127, ?	Double and add 1	255
  1, 8, 27, 64, 125, ?	Cubes	216
  132, 156, ?, 210, 240	Differences 24, 26, 28, 30	182
  8, 24, 12, 36, 18, 54, ?	×3, ÷2 alternately	27

• CSS 2018 Q8(C): Look at the series F2, __, D8, C16, B32. What number fills the blank? (5)

  1. Letters run backwards: F, E, D, C, B
  2. Numbers double: 2, 4, 8, 16, 32
  3. The blank is E4

CSS Exam Perspective

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• Most repeated area: five-part number series (2021, 2022, 2023, 2024 each had one)
• Tip: test in this order: differences, second differences, ratios, alternating rules, squares and cubes