 # How does this seemingly-trivial simplification work? In a section on inductive proofs in the book M Grimanijd 2022-07-03 Answered
How does this seemingly-trivial simplification work?
In a section on inductive proofs in the book Modelling Computing Systems: Mathematics for Computer Science (Muller, Struth) there is a simplification that is assumed to be trivial, but that I can't figure out.
It occurs in this step:
$\frac{k\left(k+1\right)\left(k+2\right)}{3}+\left(k+1\right)\left(k+2\right)\stackrel{?}{=}\frac{\left(k+1\right)\left(k+2\right)\left(k+3\right)}{3}$
How does one get from the first expression to the second?
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$\begin{array}{rl}\frac{k\left(k+1\right)\left(k+2\right)}{3}+\left(k+1\right)\left(k+2\right)& =\frac{k\left(k+1\right)\left(k+2\right)+3\left(k+1\right)\left(k+2\right)}{3}\\ & =\frac{\left(k+1\right)\left(k+2\right)\left(k+3\right)}{3}\end{array}$
###### Not exactly what you’re looking for? Riya Hansen
Hint:
$\left(k+1\right)\left(k+2\right)=\frac{3\cdot \left(k+1\right)\left(k+2\right)}{3}.$
Now combine the fractions and factor out $\left(k+1\right)\left(k+2\right)$