Question

What kind of sequence is this?

6, 30, 150, 750, ...

Answer 1

To determine the nature of the sequence 6, 30, 150, 750, we can look for a pattern or a rule governing the terms.

Let's examine the ratios between consecutive terms:

- The second term (30) divided by the first term (6) is \( \frac{30}{6} = 5 \).
- The third term (150) divided by the second term (30) is \( \frac{150}{30} = 5 \).
- The fourth term (750) divided by the third term (150) is \( \frac{750}{150} = 5 \).

From this, we can see that each term is obtained by multiplying the previous term by 5.

Therefore, the sequence can be expressed as follows:
- First term: \( a_1 = 6 \)
- Second term: \( a_2 = a_1 \times 5 = 6 \times 5 = 30 \)
- Third term: \( a_3 = a_2 \times 5 = 30 \times 5 = 150 \)
- Fourth term: \( a_4 = a_3 \times 5 = 150 \times 5 = 750 \)

In general, this sequence can be described as:
\[ a_n = 6 \times 5^{(n-1)} \]
where \( n \) is the term number.

Thus, the sequence is a geometric sequence with the first term \( 6 \) and a common ratio of \( 5 \).