jmc

Question

Let a_1, a_2, be a sequence with the following properties. (i) a_1 = 1, and (ii) a_2n=n a_n for any positive integer n. What is the value of a_2^100?

Accepted Answer

Note that align* a_2^1 &= a_2 = a_21 = 1 a_1 = 2^0 2^0 = 2^0, a_2^2 &= a_4 = a_22 = 2 a_2 = 2^1 2^0 = 2^1, a_2^3 &= a_8 = a_24 = 4 a_4 = 2^2 2^1 = 2^1+2, a_2^4 &= a_16 = a_28 = 8 a_8 = 2^3 2^1+2 = 2^1+2+3, align*and, in general, a_2^n = 2^1+2++(n-1). Because 1+2+3++(n-1) = 12n(n-1),we have a_2^100 = 2^(100)(99)/2 = 2^4950.