# Given sequences, finding the relation

#### Pranav

##### Well-known member
Problem:
Deﬁne $a_n=(1^2+2^2+ . . . +n^2)^n$ and $b_n=n^n(n!)^2$. Recall $n!$ is the product of the ﬁrst n natural numbers. Then,

(A)$a_n < b_n$ for all $n > 1$
(B)$a_n > b_n$ for all $n > 1$
(C)$a_n = b_n$ for inﬁnitely many n
(D)None of the above

Attempt:
The given sequence $a_n$ can be written as
$$a_n=\frac{n^n(n+1)^n(2n+1)^n}{6^n}$$
But I am not sure what to do now. I understand that this is a very less attempt towards the given problem but I really have no clue how someone should go about comparing these kind of sequences. Please give a few hints.

Any help is appreciated. Thanks!

Last edited:

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
Start by comparing $$\displaystyle a_2$$ and $b_2$ this will eliminate one of the first two inequalities . Then proceed by induction.

#### Pranav

##### Well-known member
Start by comparing $$\displaystyle a_2$$ and $b_2$ this will eliminate one of the first two inequalities . Then proceed by induction.
Thanks ZaidAlyafey but I seem to have figured out a better solution. Use of AM-GM inequality gives the answer in a few seconds.