Question 1192532
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Several points are marked on a line. 
Tina then marked another point between each two adjacent points on the line. 
He repeated this process a further 3 times. 
There are now 225 points marked on the line. 
How many points were marked on the line initially?
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Every time, as we start with n points on the line, we insert (n-1) points in between, and we get 

    n + (n-1) = 2n-1 points after inserting.



So, first inserting leads us from n points to 2n-1 points.



Repeating such inserting one more time leads us from 2n-1 points to 2*(2n-1)-1 = 4n-3 points.


Repeating such inserting second time leads us from 2n-1 points to 2*(4n-3)-1 = 8n-7 points.


Repeating such inserting third time leads us from 8n-7 points to 2*(8n-7)-1 = 16n-15 points.




From the given part of the problem, we have this equation

    16n - 15 = 225.



It gives

    n = {{{(225+15)/16}}} = {{{240/16}}} = 15.


<U>ANSWER</U>.  Originally, there were 15 points.


<U>CHECK</U>.  15 points give 2*15-1 = 29 pints after first inserting.

        It gives 2*29-1 =  57 points after first repeating.

        It gives 2*57-1 = 113 points after second repeating.

        It gives 2*113-1 = 225 points after third repeating.    ! Correct !
</pre>

Solved, explained and checked.