Question

The first three terms of a sequence are given. Write your answer as a decimal or whole number. Round to the nearest thousandth (if necessary).

14,7,72,...



Find the 6th Term

(1 point)
a6=

Answer 1

To find the 6th term of the sequence given the first three terms as 14, 7, and 72, we need to determine any pattern or formula that describes the sequence.

Let's analyze the relationship between the terms:

- The first term \( a_1 = 14 \)
- The second term \( a_2 = 7 \)
- The third term \( a_3 = 72 \)

To see if there's a consistent pattern, we first look for any direct operations or rules. Since no obvious arithmetic or geometric pattern emerges, we can conjecture that there might be a multiplicative relationship or a more complex rule governing the sequence.

Let's check if there is a pattern based on multiplication:

1. To get from the first term to the second term:
\[
a_2 = \frac{a_1}{2} = \frac{14}{2} = 7
\]

2. To get from the second term to the third term:
\[
a_3 = a_2 \times 10 + 2 = 7 \times 10 + 2 = 70 + 2 = 72
\]

From our initial findings, it seems each term could be generated in relation to prior terms. Let's formulate a conjectured pattern:

- From \( a_1 \) to \( a_2 \) is division,
- From \( a_2 \) to \( a_3 \) seems like a multiplication with an addition.

Now, if we investigate potential recursive formulas, let's compute the next terms similarly.

Hypothetical pattern:

Suppose the next term is determined recursively using the previous terms. We could suggest that:

- \( a_4 = a_3 \times 1 + a_2 \) (This assumes adding the previous term)
- \( a_5 = a_4 \times 1 + a_3 \)
- \( a_6 = a_5 \times 1 + a_4 \)

Calculating the terms:

1. For \( a_4 \):
\[
a_4 = 72 \times 1 + 7 = 72 + 7 = 79
\]

2. For \( a_5 \):
\[
a_5 = 79 \times 1 + 72 = 79 + 72 = 151
\]

3. For \( a_6 \):
\[
a_6 = 151 \times 1 + 79 = 151 + 79 = 230
\]

Thus, the 6th term of the sequence is:

\[
\boxed{230}
\]