Complete the following sequences by finding the pattern and filling in the blanks: a) 1, 6, 11, 16, , , b) 17, 27, 37, , , c) 2, 10, 50, , , , 31250

$$6 - 1 = 5$$ $$11 - 6 = 5$$ $$16 - 11 = 5$$

The pattern is increasing by 5.

$$16 + 5 = 21$$ $$21 + 5 = 26$$ $$26 + 5 = 31$$

The pattern is increased by 5.
The completed sequence is: $\boxed{\text{1, 6, 11, 16, 21, 26, 31}}$

$$27 - 17 = 10$$ $$37 - 27 = 10$$

The pattern is increasing by 10.

$$37 + 10 = 47$$ $$47 + 10 = 57$$ $$57 + 10 = 67$$

The pattern is increased by 10.
The completed sequence is: $\boxed{\text{17, 27, 37, 47, 57, 67}}$

Differences: $10 - 2 = 8$, $50 - 10 = 40$. Not an arithmetic sequence.
Ratios:

$$\frac{10}{2} = 5$$ $$\frac{50}{10} = 5$$

The pattern is multiplying by 5.

$$50 \times 5 = 250$$ $$250 \times 5 = 1250$$ $$1250 \times 5 = 6250$$

$$6250 \times 5 = 31250$$

This matches the given last term.
The pattern is multiplied by 5.
The completed sequence is: $\boxed{\text{2, 10, 50, 250, 1250, 6250, 31250}}$