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The positive integers $$a_1, a_2, a_3, \dots$$ form an arithmetic sequence. If $$a_1 = 10$$ and $$a_{a_2} = 100$$, find $$a_{a_{a_3}}$$.

Aug 28, 2022

#2
+2

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,

$$\displaystyle a_{a_{a_3}}=a_{a_{22}}=a_{136}=10+135.6=10+810 = 820.$$

Aug 29, 2022