1
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Find the values of n for which 1! + 2! + 3! + ... + n! is the square of a natural number.

My attempt : I tried to find the summation of factorials upto n terms. Let the summation of first n factorials be $S_n$ $$S_1=1$$ $$S_2=3$$ $$S_3=9$$ $$S_4=33$$ I could not find any generic way to find $S_n$ from this data points. Please help me.

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3
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Hint: Consider the last digit of $S_n$ for $n \geq 4$. What do you observe? Can the number ever be a perfect square?

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  • 1
    $\begingroup$ Yes, thanks a lot. I got the sense. So the answers are 1 and 3 only. $\endgroup$ – MathsLearner Jul 27 '18 at 9:47

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