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.
There are lots of ways of starting
this question. One way would be to multiply both
sides by 𝑛 plus nine factorial. Multiplying the first term by 𝑛
plus nine factorial gives us 𝑛 plus nine factorial over 𝑛 plus seven
factorial. The second term on the left-hand
side becomes 𝑛 plus nine factorial over 𝑛 plus eight factorial. As 𝑛 plus nine factorial divided
by 𝑛 plus nine factorial is equal to one, the right-hand side becomes 256.
We recall that 𝑟 factorial is
equal to 𝑟 multiplied by 𝑟 minus one factorial. This means that 𝑛 plus nine
factorial can be rewritten as 𝑛 plus nine multiplied by 𝑛 plus eight factorial or
𝑛 plus nine multiplied by 𝑛 plus eight multiplied by 𝑛 plus seven factorial. The first term therefore simplifies
to 𝑛 plus nine multiplied by 𝑛 plus eight. The second term simplifies to 𝑛
plus nine. 𝑛 plus nine multiplied by 𝑛 plus
eight plus 𝑛 plus nine is equal to 256.
We can distribute the parentheses
or expand the brackets using the FOIL method. Multiplying the first terms gives
us 𝑛 squared, the outer terms eight 𝑛, the inner terms nine 𝑛, and the last terms
72. We now have an equation 𝑛 squared
plus eight 𝑛 plus nine 𝑛 plus 72 plus 𝑛 plus nine is equal to 256. By collecting like terms, the
left-hand side simplifies to 𝑛 squared plus 18𝑛 plus 81. We can then subtract 256 from both
sides of the equation such that 𝑛 squared plus 18𝑛 minus 175 is equal to zero.
We can now factor this quadratic
expression into two sets of parentheses. The first term in each of them is
𝑛, as 𝑛 multiplied by 𝑛 is 𝑛 squared. The second terms will have a sum of
18 and a product of negative 175. 25 multiplied by seven is 175. This means that positive 25
multiplied by negative seven is negative 175. The numbers positive 25 and
negative seven also have a sum of 18. As this expression equals zero, one
of our parentheses must be equal to zero. This means that either 𝑛 is equal
to negative 25 or 𝑛 equals seven. Factorials are only defined for
nonnegative integers. This means we can discard the
solution 𝑛 equals negative 25. The value of 𝑛 that satisfies the
equation is 𝑛 equals seven. The solution set of the equation
just contains the number seven.