Example 12: Somjit noticed a repeating pattern along the border of a saree (A B C D E F …). Somjit wonders if there is a way to describe all the positions where the (i) Design A occurs, (ii) Design B occurs, and (iii) Design C occurs.
Answer
The border repeats in blocks of three: A, B, C, A, B, C, …
Design
Positions where it occurs
Formula for the nth time
A
1, 4, 7, 10, 13, …
3n – 2
B
2, 5, 8, 11, 14, …
3n – 1
C
3, 6, 9, 12, 15, …
3n
Why it happens: D, E, F in the picture are just A, B, C repeating. Every third position carries C, so C's positions are the multiples of 3. B always sits one place before C, and A two places before.
Q2.
Where would design C appear for the nth time?
Answer
Design C first appears at position 3, then 6, then 9 — the multiples of 3.
nth occurrence of Design C is at position 3n
Check it yourself: The 20th C would be at position 3 × 20 = 60.
Q3.
Similarly, find the formula that gives the position where the other Designs appear for the nth time.
Answer
B occurs at 2, 5, 8, 11, 14, … — always one less than a multiple of 3.
nth occurrence of Design B: 3n – 1 nth occurrence of Design A: 3n – 2
Why it happens: In each block of three, C sits last, B just before it and A just before B. So their positions are 3n, 3n – 1 and 3n – 2.
Q4.
Given a position number can we find out the design that appears there? Which Design appears at Position 122?
Answer
Yes, and the design at position 122 is Design B.
122 = 3 × 41 – 1 So 122 is one less than 123, a multiple of 3 Positions of the form 3n – 1 carry Design B
Why it happens: Every position is a multiple of 3 (→ C), one less than a multiple of 3 (→ B), or two less (→ A). There is no fourth possibility.
Q5.
Can the remainder obtained by dividing the position number by 3 be used for this? Observe the table below. (99 → quotient 33, remainder 0; 122 → quotient 40, remainder 2; 148 → quotient 49, remainder 1)
Answer
Yes. The remainder alone decides the design.
Remainder on dividing by 3
Form of the position
Design
0
3n
C
1
3n + 1, i.e. 3(n+1) – 2
A
2
3n + 2, i.e. 3(n+1) – 1
B
Check it yourself: Position 1 leaves remainder 1 and holds A ✔. Position 2 leaves remainder 2 and holds B ✔. Position 3 leaves remainder 0 and holds C ✔.
Q6.
Use this to find what design appears at positions 99, 122, and 148.
Answer
Position
Division by 3
Remainder
Design
99
99 = 3 × 33 + 0
0
C
122
122 = 3 × 40 + 2
2
B
148
148 = 3 × 49 + 1
1
A
99 = 3 × 33 → Design C 122 = 3 × 41 – 1 → Design B 148 = 3 × 50 – 2 → Design A