Question

Using mathematical induction, the numbers \( a_{n} \) are defined by \( a_{0} = 1,\; a_{n+1} = 3n^{2} + n + a_{n}, \; (n \ge 0) \). Then, \( a_{n} \) is equal to

• A. $n^{3}+n^{2}+1$
• B. $n^{3}-n^{2}+1$
• C. $n^{3}-n^{2}$
• D. $n^{3}+n^{2}$

Hint:

To solve induction patterns quickly, calculate the first few terms and verify options.

Answer 1

The Correct Option is B

Step 1: Generate Sequence Terms
Given $a_{0} = 1$.
$a_{1} = 3(0)^{2} + 0 + a_{0} = 0 + 1 = 1$.
$a_{2} = 3(1)^{2} + 1 + a_{1} = 3 + 1 + 1 = 5$.
Step 2: Test Option (b)
Let $P(n) = n^{3} - n^{2} + 1$.
$P(0) = 0 - 0 + 1 = 1 = a_{0}$.
$P(1) = 1 - 1 + 1 = 1 = a_{1}$.
$P(2) = 8 - 4 + 1 = 5 = a_{2}$.
Step 3: Conclusion
Since option (b) matches the calculated terms, it is the correct general form.
Final Answer: (b)