📚 SRMJEEE – Mathematics
Sequences & Series, Permutations & Combinations
Goal – Master the core ideas, formulas, and tricks that appear on the SRMJEEE exam.
Time‑budget – 2‑3 hours of focused note‑making is enough.
1. Sequences & Series
1.1 What is a Sequence?
A sequence is an ordered list of numbers.
Notation: (a_1, a_2, a_3, \dots) or ({a_n}_{n=1}^{\infty}).
1.2 Types of Sequences
| Sequence | General Term (a_n) | Sample |
|---|---|---|
| Arithmetic | (a_n = a_1 + (n-1)d) | 2, 5, 8, … |
| Geometric | (a_n = a_1 r^{,n-1}) | 3, 6, 12, … |
| Fibonacci | (F_n = F_{n-1} + F_{n-2}), (F_1 = 1, F_2 = 1) | 1, 1, 2, 3, 5, 8 … |
| Harmonic | (a_n = \frac{1}{n}) | 1, 1/2, 1/3, … |
| Quadratic | (a_n = an^2 + bn + c) | (1, 4, 9, 16, …) (square numbers) |
Tip – On the exam, the first term (a_1) and the common difference/ratio is almost always given.
1.3 Series – Sum of Terms
A series is the sum of a sequence’s terms:
[
S_n = \sum_{k=1}^{n} a_k
]
1.3.1 Arithmetic Series
[ S_n = \frac{n}{2}\bigl(2a_1 + (n-1)d\bigr) = \frac{n}{2},(a_1+a_n) ]
1.3.2 Geometric Series
- If (|r|<1) (infinite series):
[ S_\infty = \frac{a_1}{1-r} ] - Finite (n) terms:
[ S_n = a_1\frac{1-r^n}{1-r} ]
1.3.3 Harmonic Series
No simple closed form; use approximations or properties of logarithms.
1.3.4 Fibonacci Sum
[ F_1 + F_2 + \dots + F_n = F_{n+2} - 1 ]
Quick Check – Memorise the two most common series formulas (arithmetic, geometric).
2. Permutations & Combinations
2.1 Basic Counting Principles
| Concept | Formula | Example |
|---|---|---|
| Permutation (ordered arrangement) | (P(n,r)=\dfrac{n!}{(n-r)!}) | Arrange 3 out of 5 letters |
| Combination (unordered selection) | (C(n,r)=\dfrac{n!}{r!(n-r)!}) | Choose 2 fruits from 5 |
(n!) = (n \times (n-1) \times \dots \times 1)
2.2 Special Cases
| Scenario | Formula | Notes |
|---|---|---|
| All items distinct, all chosen | (n!) | Full permutation |
| Repetition allowed | (r^n) | e.g., 3‑digit codes from 0‑9 |
| Multiset permutations | (\dfrac{n!}{n_1!n_2!\dots n_k!}) | e.g., letters in MISSISSIPPI |
| Circular permutations | ((n-1)!) | For arrangements around a table |
| Permutations with restriction | Use inclusion‑exclusion or complementary counting | e.g., no two adjacent vowels |
2.3 Binomial Theorem
[ (a+b)^n = \sum_{k=0}^{n} C(n,k),a^{,n-k}b^{,k} ]
- Coefficient of (a^{,n-k}b^{,k}) is (C(n,k)).
- Useful for expanding powers and for combinatorial identities.
2.4 Pigeonhole Principle
If (n) items are put into (k) boxes and (n>k), at least one box contains (\lceil n/k\rceil) items.
3. Worked Examples
Example 1 – Arithmetic Series
Problem
Sum of first 20 terms of the sequence: 3, 7, 11, …
Solution
- (a_1 = 3), common difference (d = 4).
- (n = 20).
- (S_{20} = \dfrac{20}{2}\bigl(2\cdot3 + (20-1)\cdot4\bigr) = 10(6 + 76) = 10 \times 82 = 820).
Answer – 820