Maybe IMO should give this out

Prove that there does exist some integers $n$ such that $3^{n}+3^{n-1}+3^{n-2}+......+3^{4}+3^{3}+3^{2}+3^{1}+3^{0}$ is a perfect square, or else define the $\color{#3D99F6}{\huge{1000th}}$ smallest value of $n$ .

#NumberTheory

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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Comments

...Sorry, note edited.

Bryan Lee Shi Yang - 6 years, 3 months ago

It can be a perfect square. Smallest possible value of n will be -1.

Abhijeet Verma - 6 years, 3 months ago

I am sorry, it should be a $POSITIVE$ number.

Bryan Lee Shi Yang - 6 years, 3 months ago

You have mentioned it to be an integer. Other values will be 0,1,4

Abhijeet Verma - 6 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$