Can the remainder obtained by dividing the position number by 3 be used for this? Observe the table below. Position no. Quotient on division by 3 Remainder 99 33 0 122 40 2 148 49 1

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 ✔.

NCERT Solutions for Class 7th Maths Chapter 4 Algebraic Expressions to Describe Patterns — In-text Questions

Book page 96–97 Updated on2026-09-19

Q1.

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?

Page 96 — the repeating border of Somjit’s saree, positions 1 to 6. The six motifs are redrawn here in simplified form; the order and the repeats are as printed.

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.

Position no.	Quotient on division by 3	Remainder
99	33	0
122	40	2
148	49	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

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