Question

how many dots are in the nth term of the following sequence?
terms:
1: three dots arranged in a sort of triangular or small cluster
2: five dots, maybe a different arrangement
3: seven dots? or more? the image shows dot patterns for term 1, 2, 3, 4
4: nine dots?
choose the correct answer below.
a. ( n^2 + 2 )
b. ( n + 3 )
c. ( n + 2 )
d. ( 2n + 1 )

Explanation:

Step1: Count dots for each term

• Term 1: Let's count the dots. From the figure, Term 1 has 3 dots? Wait, no, maybe I misread. Wait, Term 1: looking at the first figure, maybe Term 1: let's check the options. Wait, maybe the terms are n=1,2,3,4. Let's list the number of dots for each term.

Wait, maybe Term 1: let's see the first diagram (Term 1) has 3 dots? No, wait the options are \( n^2 + 2 \), \( n + 3 \), \( n + 2 \), \( 2n + 1 \). Let's test n=1:

• For option A: \( 1^2 + 2 = 3 \)
• Option B: \( 1 + 3 = 4 \)
• Option C: \( 1 + 2 = 3 \)
• Option D: \( 2(1) + 1 = 3 \)

Wait, maybe Term 1: let's count the dots. Wait the first term (Term 1) has 3 dots? Wait the second term (Term 2) has 5 dots? Wait no, maybe I made a mistake. Wait let's re-examine. Wait the problem is about the number of dots in the nth term. Let's check each term:

Term 1: Let's count the dots. The first figure (Term 1) has 3 dots? Wait no, maybe the first term (n=1) has 3 dots, n=2 has 5? Wait no, let's check the options. Let's test n=1:

Option A: \( 1^2 + 2 = 3 \)

Option B: \( 1 + 3 = 4 \)

Option C: \( 1 + 2 = 3 \)

Option D: \( 2(1) + 1 = 3 \)

n=2:

Option A: \( 2^2 + 2 = 6 \)

Option B: \( 2 + 3 = 5 \)

Option C: \( 2 + 2 = 4 \)

Option D: \( 2(2) + 1 = 5 \)

n=3:

Option A: \( 3^2 + 2 = 11 \)

Option B: \( 3 + 3 = 6 \)

Option C: \( 3 + 2 = 5 \)

Option D: \( 2(3) + 1 = 7 \)

Wait, maybe I misread the figures. Wait the original problem's figures: Term 1: maybe 3 dots, Term 2: 5 dots, Term 3: 7 dots? Wait that would be 2n +1. Wait n=1: 3, n=2:5, n=3:7, n=4:9. Let's check option D: 2n +1. For n=1: 3, n=2:5, n=3:7, n=4:9. Let's see the figures:

Term 1: 3 dots (2(1)+1=3)

Term 2: 5 dots (2(2)+1=5)

Term 3: 7 dots (2(3)+1=7)

Term 4: 9 dots (2(4)+1=9)

Yes, that makes sense. So the formula is 2n +1, which is option D.

Step2: Verify with n=1,2,3,4

• n=1: \( 2(1) + 1 = 3 \) (matches the first term's dot count)
• n=2: \( 2(2) + 1 = 5 \) (matches the second term's dot count)
• n=3: \( 2(3) + 1 = 7 \) (matches the third term's dot count)
• n=4: \( 2(4) + 1 = 9 \) (matches the fourth term's dot count)

Other options:

• Option A: \( n^2 + 2 \) for n=2: 6, which doesn't match.
• Option B: \( n + 3 \) for n=2: 5, but n=3: 6, which doesn't match the third term (should be 7).
• Option C: \( n + 2 \) for n=2: 4, which doesn't match.

So the correct formula is \( 2n + 1 \), which is option D.

Answer:

D. \( 2n + 1 \)