What is the n-th term of the sequence 2, 6, 12, 20...?

Analysis

To find the $n$-th term of the sequence $2, 6, 12, 20, \dots$, we need to identify the pattern or the rule that generates each term based on its position ($n$).

Concept

A sequence can often be solved by looking at the relationship between the position $n$ (where $n = 1, 2, 3, 4, \dots$) and the value of the term.

• For $n = 1$, the term is $2$.
• For $n = 2$, the term is $6$.
• For $n = 3$, the term is $12$.
• For $n = 4$, the term is $20$.

Step-by-Step Solution

Step 1: Test the options by substituting $n = 1, 2, 3$

• Option A ($4n - 2$): • $n=1: 4(1) - 2 = 2$ (Correct) • $n=2: 4(2) - 2 = 6$ (Correct) • $n=3: 4(3) - 2 = 10$ (Incorrect, the sequence has 12)

• Option B ($2(3n - 1)$): • $n=1: 2(3(1) - 1) = 2(2) = 4$ (Incorrect, the sequence starts with 2)

• Option C ($n^2 + n$): • $n=1: (1)^2 + 1 = 1 + 1 = 2$ (Correct) • $n=2: (2)^2 + 2 = 4 + 2 = 6$ (Correct) • $n=3: (3)^2 + 3 = 9 + 3 = 12$ (Correct) • $n=4: (4)^2 + 4 = 16 + 4 = 20$ (Correct)

• Option D ($n^2 + 3n + 2$): • $n=1: (1)^2 + 3(1) + 2 = 1 + 3 + 2 = 6$ (Incorrect, the sequence starts with 2)

Step 2: Verification The formula $n^2 + n$ (which can also be written as $n(n+1)$) perfectly generates every term in the given sequence.

Conclusion

The formula that satisfies the sequence $2, 6, 12, 20$ is $n^2 + n$.

Exam Tip

When you see a sequence where the differences between terms are increasing (in this case: $6-2=4$, $12-6=6$, $20-12=8$), it is usually a quadratic sequence. If the differences are $4, 6, 8 \dots$, the second difference is constant ($2$), which confirms the formula will involve an $n^2$ term. Always test your options starting with $n=1$ to save time!