10.3 If the sequence $\left\{a_{n}\right\}$ is defined as $a_{1}=2, a_{n+1}=a_{n}+2 n(n \geqslant 1)$, then $a_{100}$ equals
(A) 9900.
(B) 9902.
(C) 9904.
(D) 10100.
(E) 10102.
(35th American High School Mathematics Examination, 1984)
[Solution] This sequence is neither an arithmetic sequence nor a geometric sequence. Subtract the preceding term from the succeeding term:
$$
\begin{array}{l}
a_{2}-a_{1}=2 \cdot 1, \\
a_{3}-a_{2}=2 \cdot 2, \\
\ldots \ldots \ldots \ldots \ldots \\
a_{100}-a_{99}=2 \cdot 99 .
\end{array}
$$

Adding both sides of these equations, we get
$$
a_{100}-a_{1}=2(1+2+\cdots+99)=99 \cdot 100=9900 \text {. }
$$

Therefore,
$$
a_{100}=9902 \text {. }
$$

So the answer is $(B)$.