MHB Find the first digit after the decimal point

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To find the first decimal digit after the decimal point in the expression $\sqrt{x^2+x+1}$ for $x=2014^{2014^{2014}}$, participants discuss the accuracy and clarity of different solutions provided. One user acknowledges a correct answer but suggests that the edited version of the solution is less straightforward than the original. The conversation emphasizes the importance of clarity in mathematical explanations. Overall, the focus remains on determining the decimal digit accurately while ensuring the solution is easy to understand. The discussion highlights the balance between correctness and clarity in mathematical communication.
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Determine the first decimal digit after the decimal point in the number $\sqrt{x^2+x+1}$ if $\large x=2014^{2014^{2014}}$
 
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anemone said:
Determine the first decimal digit after the decimal point in the number $\sqrt{x^2+x+1}$ if $\large x=2014^{2014^{2014}}$

x is too large $2014^{2014^{2014}}$

so $x^2 + x + 1= (x + 1/2)^2 + 3/4$
= $(x+1/2)^2( 1+ 3/(4(x + 1/2)^2)$
so square root = $(x+1/2) ( 1 + 3/(8(x+1/2)^2) + ...)$
the term $3/(8(x+1/2)^2)$ is extremley small so << .1
so square root is x + 1/2 or 5 is the 1st digit after decimal
 
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kaliprasad said:
x is too large $2014^{2014^{2014}}$

so $x^2 + x + 1= (x + 1/2)^2 + 3/4$
= $(x+1/2)^2( 1+ 3/(4(x + 1/2)^2)$
so square root = $(x+1/2) ( 1 + 3/(8(x+1/2)^2) + ...)$
the term $3/(8(x+1/2)^2)$ is extremley small so << .1
so square root is x + 1/2 or 5 is the 1st digit after decimal

Hey kaliprasad, thanks for participating!:) Well done! Your answer is correct... but I think this edited version of the solution isn't quite straightforward than the before edited post.:p
 
anemone said:
Hey kaliprasad, thanks for participating!:) Well done! Your answer is correct... but I think this edited version of the solution isn't quite straightforward than the before edited post.:p

this edited post is more accurate as it defines the reason. As you have made a reference to un edited post I mention the unedited post( exact words I do not remeber so in lines as below) which is highly informal

x is too large $2014^{2014^{2014}}$

so $x^2+x+1=(x+1/2)^2+3/4$ and as 3/4 is too small we can igmore so

square root =x + 1/2 so 1st digit after decimal = 5
 
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