Problem:
The sequence a1​,a2​,a3​,… satisfies a1​=19,a9​=99, and, for all n≥3,an​ is the arithmetic mean of the first n−1 terms. Find a2​.
Answer Choices:
A. 29
B. 59
C. 79
D. 99
E. 179
Solution:
For n≥3,
an​=n−1a1​+a2​+⋯+an−1​​
Thus ( n−1 ) an​=a1​+a2​+⋯+an−1​. It follows that
an+1​=na1​+a2​+⋯+an−1​+an​​=n(n−1)⋅an​+an​​=an​,
for n≥3. Since a9​=99 and a1​=19, it follows that
99=a3​=219+a2​​
and hence that a2​=179. (The sequ is 19,179,99,99,….)