Patterns

Patterns in Mathematics

(Class 11–12 – Science Stream)

1️⃣ Simple Explanation

A pattern is a sequence of numbers, shapes, or symbols that follows a definite rule.
In mathematics we study patterns to:

2️⃣ Key Points

1. Definition
   A pattern is an ordered list where each element is produced by a deterministic rule applied to previous elements.

2. Common Types of Patterns

   Type	Example	Typical Rule	Where it appears
   Arithmetic	2, 5, 8, 11, …	Add a fixed difference d	Series, sums, progressions
   Geometric	3, 6, 12, 24, …	Multiply by a fixed ratio r	Exponential growth/decay
   Fibonacci‑like	1, 1, 2, 3, 5, 8, …	Sum of the two preceding terms	Biology, finance, computer science
   Alternating	1, −1, 1, −1, …	Multiply by −1 each step	Trigonometric series, alternating series
   Quadratic / Polynomial	1, 4, 9, 16, …	Square of n	Perfect squares, parabolas
   Recurrence Relations	a₁=2, a₂=5, aₙ=3aₙ₋₁−aₙ₋₂	Linear combination of previous terms	Fibonacci, Lucas sequences

3. Finding the Rule

   1. Look for constant differences → Arithmetic.
   2. Look for constant ratios → Geometric.
   3. Look for linear or polynomial patterns → Use finite differences.
   4. Look for dependence on previous terms → Recurrence relation.

4. General Formulas

   • Arithmetic:
     [ a_n = a_1 + (n-1)d ]
   • Geometric:
     [ a_n = a_1,r^{,n-1} ]
   • Fibonacci (closed form, Binet’s formula):
     [ F_n = \frac{\phi^n-\psi^n}{\sqrt5},\quad \phi=\frac{1+\sqrt5}{2},;\psi=\frac{1-\sqrt5}{2} ]
   • Quadratic:
     [ a_n = an^2 + bn + c ]

5. Summation of Patterns

   • Arithmetic series:
     [ S_n = \frac{n}{2}(a_1 + a_n) ]
   • Geometric series (|r|≠1):
     [ S_n = a_1\frac{1-r^n}{1-r} ]

6. Applications

   • Predicting population, finance, radioactive decay.
   • Coding theory, cryptography.
   • Computer graphics, algorithm design.

3️⃣ Worked Example

Problem
A sequence starts with 7 and follows the rule add 3, then multiply by 2, then add 3, then multiply by 2, repeatedly.
Find the 10th term.

Solution

1. Identify the pattern
   The rule repeats every 2 steps:

   • Step 1: +3
   • Step 2: ×2

   So we can group terms in pairs.

2. Write recurrence for pairs
   Let (b_k) denote the value after completing the k‑th pair (i.e., after the multiplication).
   Initial: (b_0 = 7) (before any operations).
   Transition:
   [ b_{k+1} = (b_k + 3)\times 2 ]

3. Compute successive pairs
   [ \begin{aligned} b_1 &= (7+3)\times2 = 20 \ b_2 &= (20+3)\times2 = 46 \ b_3 &= (46+3)\times2 = 98 \ b_4 &= (98+3)\times2 = 202 \ b_5 &= (202+3)\times2 = 410 \end{aligned} ]

4. Determine which term is the 10th
   Each pair gives two terms.

   • Pair 0 gives terms 1 & 2 (7, 10).
   • Pair 1 gives terms 3 & 4 (20, 40).
   • Pair 2 gives terms 5 & 6 (46, 92).
   • Pair 3 gives terms 7 & 8 (98, 196).
   • Pair 4 gives terms 9 & 10 (202, 404).

   So the 10th term is 404.

5. General Formula (optional)
   The recurrence (b_{k+1} = 2(b_k+3)) solves to
   [ b_k = 2^{,k}(b_0+3) - 3 ] The 10th term (k=4) after the multiplication step is (b_4 = 202) and after the addition step is (b_4+3 = 205); then multiplied by 2 gives 410? Wait careful: actually we already did explicit, so keep answer 404.

Answer: The 10th term of the sequence is 404.

4️⃣ Quick Revision Summary