Question

question 23 (1 point)
based on the pattern, what are the next two terms of the sequence?
9, 15, 21, 27,...
○ a 162, 972
○ b 33, 972
○ c 33, 39
○ d 39, 45

question 24 (1 point)
what conjecture can you make about the fourteenth figure in this pattern?
image of figures: trapezoid, shaded horizontal rectangle, shaded vertical rectangle, then trapezoid, shaded horizontal, shaded vertical
○ a
the fourteenth figure in the pattern is image of shaded vertical rectangle
○ b
the fourteenth figure in the pattern is image of shaded horizontal rectangle
○ c
the fourteenth figure in the pattern is image of trapezoid
○ d
there is not enough information.

Explanation:

Question 23

Step1: Identify the pattern

The sequence is \(9, 15, 21, 27, \dots\). Let's find the common difference. \(15 - 9 = 6\), \(21 - 15 = 6\), \(27 - 21 = 6\). So it's an arithmetic sequence with common difference \(d = 6\).

Step2: Find the next term (5th term)

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n - 1)d\), where \(a_1 = 9\), \(d = 6\). For the 5th term (\(n = 5\)): \(a_5 = 9 + (5 - 1) \times 6 = 9 + 24 = 33\).

Step3: Find the term after that (6th term)

For the 6th term (\(n = 6\)): \(a_6 = 9 + (6 - 1) \times 6 = 9 + 30 = 39\). Wait, no, wait, wait. Wait, the 4th term is 27, so the 5th term is \(27 + 6 = 33\), 6th term is \(33 + 6 = 39\)? Wait, but option d is 39,45? Wait, no, wait my mistake. Wait \(27 + 6 = 33\) (5th term), \(33 + 6 = 39\) (6th term)? But option d is 39,45. Wait, no, wait \(27 + 6 = 33\), \(33 + 6 = 39\), \(39 + 6 = 45\). Oh! Wait, the question says "next two terms". So after 27 (4th term), the 5th term is \(27 + 6 = 33\)? No, wait 9 (1st), 15 (2nd), 21 (3rd), 27 (4th). So 5th term: \(27 + 6 = 33\)? No, 15 - 9 = 6, 21 - 15 = 6, 27 - 21 = 6. So the common difference is 6. So 4th term is 27, 5th term: \(27 + 6 = 33\), 6th term: \(33 + 6 = 39\)? But option d is 39,45. Wait, no, I think I messed up the term number. Wait 9 is term 1, 15 term 2, 21 term 3, 27 term 4. So term 5: \(27 + 6 = 33\), term 6: \(33 + 6 = 39\), term 7: \(39 + 6 = 45\). Wait, the question says "next two terms" after 27 (term 4). So next two terms are term 5 and term 6? Wait no, "next two terms" after 27 would be the 5th and 6th terms? Wait 27 is the 4th term, so next (5th) is 33, then 6th is 39? But option d is 39,45. Wait, maybe I made a mistake. Wait let's check the options. Option c: 33,39. Option d: 39,45. Wait, no, 9,15,21,27. The difference is 6. So 27 + 6 = 33 (next term), 33 + 6 = 39 (next next term). So the next two terms are 33 and 39. So option c. Wait, but let me recheck. 9,15 (9+6), 21 (15+6), 27 (21+6), so next is 27+6=33, then 33+6=39. So the next two terms are 33 and 39, which is option c.

The pattern of figures repeats every 3: trapezoid, horizontal rectangle, vertical rectangle. To find the 14th figure, divide 14 by 3. \(14 \div 3 = 4\) with a remainder of \(2\) (since \(3 \times 4 = 12\), \(14 - 12 = 2\)). This means the 14th figure is the same as the 2nd figure in the pattern. The 2nd figure is the horizontal rectangle (option b's figure).

Answer:

c. 33, 39

Question 24