Т. .
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We know: The total number of tickets is 35. Another way of saying this is the number of adults plus the number of children equals 35. i.e.:
Using the symbols provided we can write this is as:
We also know: The total money collected is $75. We know that $6 was collected for every adult or
and $1 for every child or 
. i.e.:
Using the symbols provided and adding it all up we can write this is as:
Equation 1:
Equation 2:
Using the symbols provided we can write this is as:
We also know: The total money collected is $75. We know that $6 was collected for every adult or
Using the symbols provided and adding it all up we can write this is as:
Equation 1:
Equation 2:
Let
where we assume |r| < 1. Multiplying on both sides by r gives
and subtracting this from gives
As n → ∞, the exponential term will converge to 0, and the partial sums will converge to
Now, we're given
We must have |r| < 1 since both sums converge, so
Solving for r by substitution, we have
Recalling the difference of squares identity, we have
We've already confirmed r ≠ 1, so we can simplify this to
It follows that
and so the sum we want is
which doesn't appear to be either of the given answer choices. Are you sure there isn't a typo somewhere?