Patterns in Mathematics (Set 6)

1. An astronomer observes a meteor shower where the number of meteors per hour follows a specific pattern. In the $1^{st}$ hour, there are $7$ meteors. In the $2^{nd}$ hour, $13$ meteors. In the $3^{rd}$ hour, $19$ meteors. If this linear pattern continues, how many meteors will be seen in the $n^{th}$ hour?

$7n$

$6n + 7$

$6n + 1$

$n + 6$

2. A robotics engineer is programming a droid to move in a square spiral. The first side length is $2$ units, the second is $4$ units, the third is $6$ units, and so on. What is the total distance covered by the droid after completing the $15^{th}$ side?

$225$ units

$240$ units

$480$ units

$210$ units

3. Consider a pattern of matchstick squares arranged in a row. If $1$ square requires $4$ sticks, $2$ squares (sharing one side) require $7$ sticks, and $3$ squares require $10$ sticks. How many sticks are needed to form a row of $100$ squares?

$400$

$304$

$299$

$301$

4. A digital sequence is defined by the rule: $Term = n^2 + (n-1)$. What is the $12^{th}$ term of this sequence?

$155$

$143$

$154$

$133$

5. In a triangular number pattern, the $n^{th}$ number is given by $T_n = \frac{n(n+1)}{2}$. If a specific triangular number is $120$, what is the value of $n$?

$12$

$14$

$15$

$16$

6. A pattern of hexagonal floor tiles is laid out. Each hexagon shares sides with adjacent hexagons. If $1$ hexagon has a perimeter of $6$ units (side = $1$), and a chain of $2$ hexagons has a perimeter of $10$ units, what is the perimeter of a chain of $n$ hexagons?

$6n$

$4n + 2$

$5n + 1$

$4n - 2$

7. A sequence of numbers is generated where each term is twice the previous term minus $3$. If the first term is $5$, what is the $5^{th}$ term?

$17$

$31$

$37$

$35$

8. Which algebraic expression correctly represents the sum of three consecutive natural numbers where the middle number is $m$?

$3m$

$3m + 3$

$3m - 3$

$m^3$

9. A pattern follows the rule: Add $5$ then multiply by $2$. If we start with $x$ and perform this operation twice, what is the final expression?

$2x + 10$

$4x + 20$

$4x + 30$

$2x + 15$

10. An ancient staircase is built such that the $1^{st}$ step is made of $1$ block, the $2^{nd}$ step of $3$ blocks, the $3^{rd}$ step of $6$ blocks, and the $4^{th}$ step of $10$ blocks. How many blocks are needed just for the $10^{th}$ step?

$45$

$55$

$66$

$100$