Grade 6 : Mathematics (Ganita Prakash) - Patterns in Mathematics (Set 4) | Concept of Maths

Question 1

1. A chain of regular hexagons is made using matchsticks. One hexagon uses 6 sticks. When two hexagons share one side, they use 11 sticks. Three hexagons sharing sides in a row use 16 sticks. How many matchsticks are needed to form a chain of 10 hexagons?

A) 50

B) 51

C) 60

D) 56

Answer

Correct Answer: B

Solution: Let $n$ be the number of hexagons. For $n=1$, sticks = $6$ For $n=2$, sticks = $11$ (which is $6 + 5$) For $n=3$, sticks = $16$ (which is $6 + 5 + 5$) The pattern follows the rule: $5n + 1$. For $n=10$: $$5 \times 10 + 1 = 50 + 1 = 51$$ So, 51 matchsticks are needed.

Question 2

2. A "T" shape is constructed from unit squares. Stage 1 uses 4 squares. Stage 2 uses 7 squares, and Stage 3 uses 10 squares. Each subsequent stage adds 3 squares to the vertical tail of the "T". How many squares are in Stage 25?

A) 75

B) 76

C) 79

D) 81

Answer

Correct Answer: B

Solution: Let $n$ be the stage number. Squares at Stage 1 = $4$ Squares at Stage 2 = $4 + 3 = 7$ Squares at Stage 3 = $7 + 3 = 10$ The rule for $n$ stages is $3n + 1$. For $n=25$: $$3 \times 25 + 1 = 75 + 1 = 76$$

Question 3

3. A pattern of dots forms a rectangular array. The 1st array is $1 \times 2$ (2 dots), the 2nd is $2 \times 3$ (6 dots), and the 3rd is $3 \times 4$ (12 dots). If this pattern continues, how many dots will be in the 12th array?

A) 132

B) 144

C) 156

D) 120

Question 4

4. In a number sequence, the $n$-th term is given by the algebraic expression $7n - 5$. If the value of a certain term is 142, what is the position ($n$) of that term in the sequence?

A) 20

B) 21

C) 19

D) 22

Answer

Correct Answer: B

Solution: We are given the rule $7n - 5$ and the value 142. Set up the equation: $$7n - 5 = 142$$ $$7n = 142 + 5$$ $$7n = 147$$ $$n = \frac{147}{7} = 21$$ The term is at the 21st position.

Question 5

5. A sequence starts with the number 2. Each subsequent term is found by squaring the previous term and then subtracting 1. The sequence starts: $2, 3, 8, \dots$. What is the 4th term?

A) 15

B) 63

C) 64

D) 56

Answer

Correct Answer: B

Solution: Term 1 = $2$ Term 2 = $2^2 - 1 = 4 - 1 = 3$ Term 3 = $3^2 - 1 = 9 - 1 = 8$ Term 4 = $8^2 - 1 = 64 - 1 = 63$

Question 6

6. Triangular numbers are formed by stacking blocks in layers. Layer 1 has 1 block, Layer 2 has 3 total blocks (1+2), Layer 3 has 6 blocks (1+2+3). How many total blocks are needed to build a stack with 10 layers?

A) 45

B) 50

C) 55

D) 60

Answer

Correct Answer: C

Solution: The sum of the first $n$ natural numbers is given by the formula: $$S = \frac{n(n+1)}{2}$$ For Layer 10 ($n=10$): $$S = \frac{10 \times (10+1)}{2} = \frac{10 \times 11}{2} = 5 \times 11 = 55$$

Question 7

7. Consider a square grid of $n \times n$ small squares. If you color only the squares that touch the outer boundary, you find that a $3 \times 3$ grid has 8 border squares and a $4 \times 4$ grid has 12 border squares. How many border squares are there for a $20 \times 20$ grid?

A) 72

B) 76

C) 80

D) 400

Answer

Correct Answer: B

Solution: Method 1: The total squares are $n^2$. The inner (uncolored) squares form an $(n-2) \times (n-2)$ grid. Rule: $n^2 - (n-2)^2$. For $n=20$: $$20^2 - 18^2 = 400 - 324 = 76$$ Method 2: Rule is $4n - 4$. $$4(20) - 4 = 80 - 4 = 76$$

Question 8

8. A pattern follows the rule: $Term = n^2 + 2n$. What is the difference between the 10th term and the 9th term of this sequence?

A) 21

B) 19

C) 23

D) 20

Answer

Correct Answer: A

Solution: Calculate the 10th term ($n=10$): $$10^2 + 2(10) = 100 + 20 = 120$$ Calculate the 9th term ($n=9$): $$9^2 + 2(9) = 81 + 18 = 99$$ Difference: $$120 - 99 = 21$$

Question 9

9. A series of houses is drawn using toothpicks. To make 1 house, 5 toothpicks are needed. To make 2 houses connected side-by-side, 9 toothpicks are needed. To make 3 houses, 13 toothpicks are needed. How many toothpicks are needed for 15 connected houses?

A) 61

B) 57

C) 59

D) 60

Answer

Correct Answer: A

Solution: Pattern: $n=1 \rightarrow 5, n=2 \rightarrow 9, n=3 \rightarrow 13$. Each house adds 4 toothpicks after the first one. Rule: $4n + 1$. For $n=15$: $$4 \times 15 + 1 = 60 + 1 = 61$$

Question 10

10. It is a known pattern that the sum of the first $n$ odd numbers is equal to $n^2$. For example, $1+3+5 = 3^2 = 9$. What is the sum of all odd numbers from 1 to 39?

A) 400

B) 441

C) 380

D) 1600

Answer

Correct Answer: A

Solution: First, find how many odd numbers are there from 1 to 39. The $n$-th odd number is $2n - 1$. $$2n - 1 = 39 \Rightarrow 2n = 40 \Rightarrow n = 20$$ So there are 20 terms. Sum = $n^2 = 20^2 = 400$.

Accepted Answer

Correct Answer: C

Solution: The $n$-th array has dimensions $n \times (n+1)$. Number of dots = $n(n+1)$. For $n=12$: $$12 \times (12 + 1) = 12 \times 13 = 156$$

Grade 6 : Mathematics (Ganita Prakash) - Patterns in Mathematics (Set 4) | Concept of Maths