Welcome to our community

Be a part of something great, join today!

Find the first digit after the decimal point

  • Thread starter
  • Admin
  • #1

anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,756
Determine the first decimal digit after the decimal point in the number $\sqrt{x^2+x+1}$ if $\large x=2014^{2014^{2014}}$
 

kaliprasad

Well-known member
Mar 31, 2013
1,322
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
 
Last edited:
  • Thread starter
  • Admin
  • #3

anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,756
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
 

kaliprasad

Well-known member
Mar 31, 2013
1,322
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
 
Last edited: