Question 3158
To solve this problem work from the last fisherman forward. How many fish were in the basket when the last fisherman got there?<br>

Since the last fisherman was able to divide the number in the basket by 3 with a remainder of 1, the number of fish must have been:<br>

{{{N = 3x + 1}}}<br>

Where x is the number of fish taken by the last fisherman.<br>
The second fisherman removed 1 then took 1/3 of the fish leaving 2/3 for the last fisherman. To construct that number we have to undo this operation, that is we multiply N by 3/2 then add 1, or:<br>

{{{M = (3/2)(3x+1)+ 1 = (9/2)x + (5/2)}}}

Likewise, the first fisherman removed 1 and left 2/3 in the basket for the second fisherman so<br>

{{{T = (3/2)M + 1 = (3/2)((9/2)x + (5/2)) + 1 = (27/4)x + 19/4}}}<br>

If we try n = 0 we cannot construct a valid number because T would not be a whole number. Likewise, n = 1 and n = 2 do not work, but n = 3 does. So, with x = 3, the number of fish in the basket when the second fisherman took his turn was:<br>
{{{M = (9/2)x + (5/2) = 16}}},<br>
which means he took {{{(16-1)/3 = 5}}} fish. When the first fisherman gets to the basket there are:<br>
{{{T = (3/2)(16)+1 = 25}}}
The fishermen caught 25 fish.<br>
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To construct the total number of fish, we created an equation for T in terms of x, which is what part b of the question wants. So,<br>
{{{T = (3/2)M + 1 = (3/2)((9/2)x + (5/2)) + 1 = (27/4)x + 19/4}}}<br>
see above
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Now, to find the next numbers in the sequence start with values of n greater than 3. You should quickly realize that even numbers won't work. To give you an idea of how long this will take, the value of n is 11 which makes T = 79.