Question 1198416
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An inheritance is split among 5 brothers. 
The first receives half of the inheritance plus $1. 
The second receives half of the remainder plus $2. 
The third receives half of the remainder plus $3. 
The fourth receives half of the remainder plus $4. 
The last brother receives {{{highlight(highlight(the_remaining))}}} $500. 
What is the total amount of the inheritance?
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<pre>
Let the brothers be B1,  B2,  B3,  B4,  and  B5.

Let  b[0]  be the entire inheritance.


Brother B1 got half of inheritance + $1,  or  b[0]/2 + 1 dollars.  The remainder is  r[1] = b[0]/2 - 1.    (1)


Brother B2 got half of the remainder r[1] plus $2, or r[1]/2 + 2.  The remainder is  r[2] = r[1]/2 - 2.    (2)


Brother B3 got half of the remainder r[2] plus $3, or r[2]/2 + 3.  The remainder is  r[3] = r[2]/2 - 3.    (3)


Brother B4 got half of the remainder r[3] plus $4, or r[3]/2 + 4.  The remainder is  r[4] = r[3]/2 - 4.    (4)


r[4] is what brother B5 got:  r[4] = 500 = r[3]/2 - 4.

From it,   we have  r[3]/2 = 500 + 4 = 504,  which gives  r[3] = 2*504 = 1008.


    +----------------------------------------------------------------------+
    |   Now I will solve equations (3), (2), (1)  in this backward order.  |
    +----------------------------------------------------------------------+


From (3),  we have  r[2]/2 - 3 = r[3] = 1008,  or  r[2]/2 = 1008+3 = 1011,  which gives  r[2] = 2*1011 = 2022.


From (2),  we have  r[1]/2 - 2 = r[2] = 2022,  or  r[1]/2 = 2022+2 = 2024,  which gives  r[1] = 2*2024 = 4048.


From (1),  we have  b[0]/2 - 1 = r[1] = 4048,  or  b[0]/2 = 4048+1 = 4049,  which gives  b[0] = 2*4049 = 8098.


<U>ANSWER</U>.  The whole inheritance was 8098 dollars.
</pre>

Solved.


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The method which I used is called a "backward solution method".


On the way forward I establish the equations.


On the way back I solve these equations, one after another.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Nice method, isn't it ?