Sequence and Series for CA Foundation
Arithmetic and geometric progressions, their nth terms and sums, means, and the sums of natural numbers, squares and cubes.
Quantitative AptitudePaper 3, Business Mathematics12 MCQs2 lessons2 flashcards3 memory tricks
Practise Sequence and Series free
What you'll learn
- Arithmetic progressions
Find terms and sums of an AP. - Geometric progressions
Find terms and sums of a GP, including infinite GPs.
Key formulas
| nth term | tₙ = a + (n − 1)d |
|---|---|
| Sum of n terms | Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l) |
| Three terms in AP | a, b, c with 2b = a + c |
| Arithmetic mean of a, b | (a + b)/2 |
| 1 + 2 + … + n | n(n + 1)/2 |
| 1² + 2² + … + n² | n(n + 1)(2n + 1)/6 |
| 1³ + 2³ + … + n³ | [n(n + 1)/2]² |
| nth term | tₙ = arⁿ⁻¹ |
Concepts explained
Arithmetic progression
An AP has a constant difference d between terms: tₙ = a + (n − 1)d and Sₙ = n/2 [2a + (n − 1)d].
What it means
- Middle of three terms is their average: 2b = a + c.
- Sum = number of terms × average of first and last.
- Take three terms as a − d, a, a + d to simplify.
- Find which term equals x: solve a + (n − 1)d = x.
- 3, 7, 11, …
- 3 + 9 × 4
Answer: 39
- 5, 9, 13, … = 101
- (101 − 5)/4 + 1
Answer: 25th
Related terms
- Arithmetic mean: (a + b)/2
- nth term from the end: l − (n − 1)d
Exam tip: Number of steps is one less than the term number.
Geometric progression
A GP has a constant ratio r: tₙ = arⁿ⁻¹, Sₙ = a(rⁿ − 1)/(r − 1), and S∞ = a/(1 − r) when |r| < 1.
What it means
- Middle of three terms in GP: b² = ac (b is the geometric mean).
- Take three terms as a/r, a, ar; product = a³.
- AM ≥ GM ≥ HM for positive numbers.
- An infinite GP converges only when |r| < 1.
- 2, 6, 18, …
- 2 × 3⁵
Answer: 486
- 1 + ½ + ¼ + …
- 1/(1 − ½)
Answer: 2
Related terms
- Compound interest: amounts form a GP
- Harmonic mean: 2ab/(a + b) for two numbers
Exam tip: For doubling series, sum of n terms = next term − first term.
Sums of natural numbers, squares and cubes
Three standard sums: Σn = n(n + 1)/2, Σn² = n(n + 1)(2n + 1)/6, Σn³ = [n(n + 1)/2]².
What it means
- Σn³ = (Σn)².
- For n = 10: 55, 385, 3025.
- Sum of first n odd numbers = n².
- Sum of first n even numbers = n(n + 1).
- n = 10
- 10 × 11 × 21 / 6
Answer: 385
- n = 5
- 15²
Answer: 225
Related terms
- Gauss's trick: pair first and last terms
- Odd numbers: 1 + 3 + 5 + … + (2n − 1) = n²
Exam tip: Learn the n = 10 values; many questions can be checked against them.
Memory tricks
AP: common difference d; GP: common ratio r.
Σn = 55, Σn² = 385, Σn³ = 3025 for n = 10; Σn³ = (Σn)².
S∞ = a/(1 − r) when |r| < 1.
Try 3 CA Foundation-style questions
Q1. The sum of the first 20 natural numbers is
- 200
- 210
- 420
- 190
Show answer
Answer: (2) 210
n(n + 1)/2 = 20 × 21 / 2 = 210.
Q2. The geometric mean of 4 and 16 is
- 10
- 8
- 12
- 64
Show answer
Answer: (2) 8
GM = √(4 × 16) = √64 = 8. (Their arithmetic mean is 10.)
Q3. The sum of the infinite GP 1 + ½ + ¼ + … is
- 2
- 1
- ∞
- 1.5
Show answer
Answer: (1) 2
S∞ = a/(1 − r) = 1/(1 − ½) = 2, valid because |r| < 1.
Practise all 12 questions Free · explanations for every option · see where you go wrong