SOLUTION: 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 poin
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-> SOLUTION: 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 poin
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Question 1192532: 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? Answer by ikleyn(52832) (Show Source):
You can put this solution on YOUR website! .
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 = = = 15.
ANSWER. Originally, there were 15 points.
CHECK. 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 !
