Insert three fractions between $\frac1/4$ and $\frac1/2$ so the five fractions form an arithmetic sequence. What is the sum, expressed as a common fraction, of these three new fractions?
Do you mean this problem:
Insert three fractions between $\frac14$ and $\frac12$ so the five fractions form an arithmetic sequence. What is the sum, expressed as a common fraction, of these three new fractions?
If you do then the answer is 9/8
this is the reasoning:
The term in the middle is the arithmetic mean of $\dfrac{1}{4}$ and $\dfrac{1}{2}$, or $\dfrac{3}{8}$, so the arithmetic sequence looks like $$\dfrac{1}{4}, \underline{\qquad} , \dfrac{3}{8}, \underline{\qquad} , \dfrac{1}{2}.$$The fraction $\dfrac{3}{8}$ is also the arithmetic mean of the two missing terms, and thus $\dfrac{3}{8}$ is the average of the $3$ new fractions. So, their sum is $3\cdot\dfrac{3}{8}=\boxed{\dfrac{9}{8}}$.
Do you mean this problem:
Insert three fractions between $\frac14$ and $\frac12$ so the five fractions form an arithmetic sequence. What is the sum, expressed as a common fraction, of these three new fractions?
If you do then the answer is 9/8
this is the reasoning:
The term in the middle is the arithmetic mean of $\dfrac{1}{4}$ and $\dfrac{1}{2}$, or $\dfrac{3}{8}$, so the arithmetic sequence looks like $$\dfrac{1}{4}, \underline{\qquad} , \dfrac{3}{8}, \underline{\qquad} , \dfrac{1}{2}.$$The fraction $\dfrac{3}{8}$ is also the arithmetic mean of the two missing terms, and thus $\dfrac{3}{8}$ is the average of the $3$ new fractions. So, their sum is $3\cdot\dfrac{3}{8}=\boxed{\dfrac{9}{8}}$.
