how do i get the answer to this question?---**Question Stem:**
Suppose that $a_1 = 2$ and the sequence $(a_n)$ satisfies the recurrence relation
$$ \frac{a_n - 1}{n - 1} = \frac{a_{n-1} + 1}{(n - 1) + 1} $$
for all $n \geq 2$. What is the greatest integer less than or equal to
$$ \sum_{n=1}^{100} a_n^2? $$
**Options:**
(A) 338,550
(B) 338,551
(C) 338,552
(D) 338,553
(E) 338,554
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Let's solve this step by step. We have a sequence with a₁ = 2 and a recurrence relation. First, we'll rearrange the recurrence relation to find a pattern. Cross-multiplying gives us n times a_n minus 1 equals n minus 1 times a_n minus 1 plus 1. This simplifies to n times a_n equals n minus 1 times a_n minus 1 plus 2n minus 1.
Now let's verify our proposed formula a_n equals n plus 1 over n. First, we check the initial condition: a_1 equals 1 plus 1 over 1, which equals 2. This matches perfectly. Next, we verify the recurrence relation. The left side becomes n squared minus n plus 1, divided by n times n minus 1. The right side simplifies to the same expression. Since both sides are equal, our formula is correct.
Now we compute the sum of a_n squared from n equals 1 to 100. Since a_n equals n plus 1 over n, we have a_n squared equals n squared plus 2 plus 1 over n squared. We can separate this into three sums. The sum of squares from 1 to 100 equals 338,350 using the formula. The sum of the constant 2 is 200. The sum of 1 over n squared is approximately 1.635. Adding these together gives 338,551.635, so the greatest integer less than or equal to this sum is 338,551.
Let's summarize our solution. We found that a_n equals n plus 1 over n by analyzing the recurrence relation. Then we computed a_n squared equals n squared plus 2 plus 1 over n squared. The sum from n equals 1 to 100 gives us 338,551.635. Taking the greatest integer less than or equal to this value gives us 338,551, which corresponds to option B in the multiple choice answers.
To summarize our problem-solving approach: we started by rearranging the recurrence relation algebraically, then calculated the first few terms to identify a pattern. We proposed the closed-form formula a_n equals n plus 1 over n, verified it satisfies both the initial condition and recurrence relation, and finally applied it to compute the required sum. This systematic method of pattern recognition and verification is essential for solving complex sequence problems. The final answer is 338,551.