Copy these linear sequences and find the missing numbers: 2, 3 2/7, __, 5 6/7, __, 8 3/7, 9 5/7.

Steps to Solve

1. Identify the pattern in the sequence

To find the missing numbers, first observe the progression between the given numbers. The sequence starts with $2$ and ends with $9 \frac{5}{7}$.

2. Convert all numbers to improper fractions

Converting the mixed numbers to improper fractions makes calculations easier:

• $2 = \frac{14}{7}$
• $3 \frac{2}{7} = \frac{23}{7}$
• $5 \frac{6}{7} = \frac{41}{7}$
• $8 \frac{3}{7} = \frac{59}{7}$
• $9 \frac{5}{7} = \frac{68}{7}$

The sequence now is: $$ \frac{14}{7}, \frac{23}{7}, _, \frac{41}{7}, _, \frac{59}{7}, \frac{68}{7} $$

3. Determine the step size

To find the differences:

• From $2$ to $3 \frac{2}{7}$: $$ \frac{23}{7} - \frac{14}{7} = \frac{9}{7} $$
• From $3 \frac{2}{7}$ to $5 \frac{6}{7}$ (next number): $$ \frac{41}{7} - \frac{23}{7} = \frac{18}{7} $$
• This pattern suggests the differences increase by an additional $9/7$ each time.

Continuing the differences:

• From $5 \frac{6}{7}$ to $8 \frac{3}{7}$: $$ \frac{59}{7} - \frac{41}{7} = \frac{18}{7} $$
• From $8 \frac{3}{7}$ to $9 \frac{5}{7}$: $$ \frac{68}{7} - \frac{59}{7} = \frac{9}{7} $$

Continuing back, to fill in the blanks: 4. Calculate missing values

Following the pattern, we find:

• The first missing number after $3 \frac{2}{7}$: $$ 23/7 + 9/7 = \frac{32}{7} = 4 \frac{4}{7} $$

• The second missing number before $5 \frac{6}{7}$: $$ 41/7 - 9/7 = \frac{32}{7} $$

Now let's place the known values: $$ 2, 3 \frac{2}{7}, 4 \frac{4}{7}, 5 \frac{6}{7}, 6 \frac{6}{7}, 8 \frac{3}{7}, 9 \frac{5}{7} $$

5. Final sequence

So the completed linear sequence is: $$ 2, 3 \frac{2}{7}, 4 \frac{4}{7}, 5 \frac{6}{7}, 6 \frac{6}{7}, 8 \frac{3}{7}, 9 \frac{5}{7} $$