The $n^{th}$ terms of the series $3 + 7 + 13 + 21 + \dots $ is
- $4n - 1$
-
$n^n + 2n$
- $n^2 + n + 1$
- $n^2 + 2$
Reveal answer
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C
Correct answer
Explanation
The sequence is 3, 7, 13, 21. The differences are 4, 6, 8, which form an arithmetic progression. The n-th term is a quadratic of the form an^2 + bn + c. Solving for n=1, 2, 3 gives 3, 7, 13, which yields n^2 + n + 1.
AI explanation
The differences between consecutive terms are $7 - 3 = 4$, $13 - 7 = 6$, and $21 - 13 = 8$. Because the second differences are constant at 2, the formula for the $n$-th term is a quadratic expression of the form $an^2 + bn + c$. By substituting the first three terms, we find the formula is $n^2 + n + 1$. The result is $n^2 + n + 1$.