2. A sequence $a_{1}, \ldots, a_{100}$ of real numbers is such that the arithmetic mean of two consecutive terms is always equal to the index of the second term (for example, $\frac{a_{4}+a_{5}}{2}=5$); what is the sum of the 100 numbers in the sequence?
(A) 2550
(B) 5050
(C) 5100
(D) 10100
(E) It cannot be determined: it depends on $a_{1}$.
2. The answer is (C). Let $S$ be the sum of the 100 terms of the sequence. We have:

$$
\frac{S}{2}=\frac{1}{2}\left(a_{1}+a_{2}+\ldots+a_{99}+a_{100}\right)=\frac{a_{1}+a_{2}}{2}+\frac{a_{3}+a_{4}}{2}+\ldots+\frac{a_{99}+a_{100}}{2} .
$$

Notice that each of the addends on the far right is the arithmetic mean between two consecutive terms of the sequence, which we know is equal to the index of the second term. Therefore, we get

$$
\frac{S}{2}=2+4+\ldots+100=2(1+2+\ldots+50)
$$

and recalling that the sum of the first $n$ integers is $\frac{n(n+1)}{2}$, we conclude that

$$
S=4(1+2+\ldots+50)=4 \cdot \frac{50 \cdot 51}{2}=5100 .
$$