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#### Pranav

##### Well-known member

- Nov 4, 2013

- 428

**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!

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