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 \[\color{BLUE}{\huge{1000th}}\] smallest value of \(n\).

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TopNewest...Sorry, note edited. – Bryan Lee Shi Yang · 2 years ago

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You have mentioned it to be an integer. Other values will be 0,1,4 – Abhijeet Verma · 2 years ago

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I am sorry, it should be a \(POSITIVE\) number. – Bryan Lee Shi Yang · 2 years ago

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It can be a perfect square. Smallest possible value of n will be -1. – Abhijeet Verma · 2 years ago

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