Let 729,81,9,1, be a sequence and P _ n denote the product of the first n terms of this sequence. If 2 _ n=1 ⁴⁰ ( P _ n )^ 1 n = 3^ -1 3^ and gcd ( , )=1 , then + is equal to

Step-by-step solution

The sequence is 3^6, 3^4, 3^2, 3^0, 3⁻², so a_n = 3^ 8-2n . P_n = _ k=1 ^n 3^ 8-2k = 3^ _ k=1 ^n(8-2k) = 3^ 8n - n(n+1) = 3^ n(7-n) (P_n)^ 1/n = 3^ 7-n 2 _ n=1 ⁴⁰ 3^ 7-n = 2(3^6+3^5+ +3⁻³³) = 2 3^6 1-3⁻⁴⁰ 1-1/3 = 3^7 - 3⁻³³ = 3⁴⁰-1 3³³ So = 40, ; = 33 . Since (40,33)=1 , we get + = 73 .