Question

how to show the sequence {yn}, with
yn=1/((2^2)^n) is at least superlinear ?

Answer 1

for n=1 1^2=1*(1+1)(2*1+1)/6

1=1

let n=k 1^2+2^2+...+k^2=k(k+1)(2k+1)/6

we want to prove for n=k+1

1^2+2^2+...+k^2+(k+1)^2=(k+1)(k+2)(2k+…

but 1^2+2^2+...+k^2=k(k+1)(2k+1)/6

then we must prove

k(k+1)(2k+1)/6+(k+1)^2=(k+1)(k+2)(2k+3…

divide by k+1 and maltiply by 6


k(2k+1)+6(k+1)=(k+2)(2k+3)

2k^2+k+6k+6=2k^2+3k+4k+6

2k^2+7k+6=2k^2+7k+6

true for all k

Answer 2

replace all k by Y