+99 The positive integers \(a_1, a_2, a_3, \dots\) form an arithmetic sequence. If \(a_1 = 10\) and \(a_{a_2} = 100\), find \(a_{a_{a_3}}\).
Let the common difference be d, then
\(\displaystyle a_{1}=10, \\ a_{2} = 10 + d, \\ a_{3}=10 + 2d, \\ \dots \\ a_{n}=10 + (n-1)d, \\ \text{etc.} \)
\(\displaystyle a_{a_{2}}=a_{10+d}=10 +(9+d)d=d^{2}+9d+10 = 100, \\ d^{2}+9d-90=0,\\ (d-6)(d+15)=0,\\ d=6, (\text{or } d=-15). \)
So then, for example, and in anticipation,
\(\displaystyle a_{3} = 10 + (3-1).6 = 22, \\ a_{22}=10+(22-1).6= 136.\)
Finally,
\(\displaystyle a_{a_{a_3}}=a_{a_{22}}=a_{136}=10+135.6=10+810 = 820.\)
