## Thinking Mathematically (6th Edition)

$\frac{5}{6}$ yes
In each item in the sequence, the previous sum is added to a fraction whose numerator s 1 and whose denominator is a product where each factor is one greater than the factors in the final fraction of the previous sum. Since the last fraction in the given sequence is $\frac{1}{4\times5}$ the last fraction in the next item in the sequence is $\frac{1}{5\times6}$. $\underline{Inductive\ Reasoning}$ Examining each item in the sequence you can see that the result of the computation is a fraction whose numerator is the first factor in the denominator of the last fraction and whose denominator is the second factor in the denominator of the last fraction. Since the last fraction in the next computation is the sequence is $\frac{1}{5\times6}$, we can induce that the result of the computation is $\frac{5}{6}$. $\underline{Check\ Conjecture}$ The next computation in the sequence is: $\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+\frac{1}{4\times5}+\frac{1}{5\times6}$ Substituting from the previous sequence we know this is equivalent to $\frac{4}{5}+\frac{1}{5\times6}=\frac{24}{30}+\frac{1}{5\times6}=\frac{25}{30}=\frac{5}{6}$ This result is the same as the answer arrived at by deductive reasoning.