Find the solution set of one over
𝑛 plus seven factorial plus one over 𝑛 plus eight factorial is equal to 256 over
𝑛 plus nine factorial.
In order to answer this question,
we begin by recalling that for any integer 𝑛 greater than or equal to one, then 𝑛
factorial is equal to 𝑛 multiplied by 𝑛 minus one factorial. The fact that 𝑛 must be greater
than or equal to one will be important when finding the solution set.
When dealing with a problem where
the sum of two fractions is equal to another fraction, it is often useful to try and
eliminate the denominators first. In this question, we will multiply
all three terms by 𝑛 plus nine factorial. This gives us 𝑛 plus nine
factorial over 𝑛 plus seven factorial plus 𝑛 plus nine factorial over 𝑛 plus
eight factorial is equal to 256 multiplied by 𝑛 plus nine factorial over 𝑛 plus
nine factorial. On the right-hand side of our
equation, we can cancel the common factor of 𝑛 plus nine factorial, leaving us with
Using the fact that 𝑛 factorial is
equal to 𝑛 multiplied by 𝑛 minus one factorial and it is also equal to 𝑛
multiplied by 𝑛 minus one multiplied by 𝑛 minus two factorial, we can rewrite the
left-hand side of our equation as shown. In the first term, we can cancel a
factor of 𝑛 plus seven factorial. And in the second term, we cancel a
factor of 𝑛 plus eight factorial. We now have an equation that no
longer contains any fractions. 𝑛 plus nine multiplied by 𝑛 plus
eight plus 𝑛 plus nine is equal to 256.
We can simplify the left-hand side
either by taking out a factor of 𝑛 plus nine or by using the FOIL method to
distribute our parentheses. 𝑛 plus nine multiplied by 𝑛 plus
eight is equal to 𝑛 squared plus eight 𝑛 plus nine 𝑛 plus 72. And when we add 𝑛 plus nine to
this, we get an answer of 256. We can then group or collect the
like terms on the left-hand side. This gives us 𝑛 squared plus
18𝑛. And when we subtract 256 from both
sides, we have negative 175. We now have a quadratic equation 𝑛
squared plus 18𝑛 minus 175 is equal to zero.
Our next step is to factor the
expression on the left-hand side into two sets of parentheses. Since the coefficient of 𝑛 squared
is equal to one, we know that the first term in each of them will be 𝑛. And the second terms will have a
sum of positive 18 and a product of negative 175. One factor pair of 175 is 25 and
seven. This means that multiplying 25 by
negative seven gives us negative 175. And since 25 plus negative seven is
18, our two sets of parentheses are 𝑛 plus 25 and 𝑛 minus seven.
As the product of 𝑛 plus 25 and 𝑛
minus seven equals zero, then either 𝑛 plus 25 equals zero or 𝑛 minus seven equals
zero. This gives us two possible
solutions 𝑛 equals negative 25 and 𝑛 equals seven. As already mentioned, we know that
𝑛 must be greater than or equal to one, as factorials are only defined for
nonnegative integers. The value of 𝑛 that satisfies the
equation is therefore equal to seven. And the solution set of the
equation one over 𝑛 plus seven factorial plus one over 𝑛 plus eight factorial is
equal to 256 over 𝑛 plus nine factorial contains the number seven.