Question Video: Evaluating Factorials to Find the Values of Unknowns Mathematics

Find the solution set of (1/(𝑛 + 7)!) + (1/(𝑛 + 8)!) = 256/(𝑛 + 9)!.

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Video Transcript

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.

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