Question

in a sequence of numbers,
a3=0,a4=4,a5=8,a6=12,a7=16
based on this information, which equation can be used to find the n^th term in the sequence, an?
a an=-4n-6
b an=-4n-12
c an=4n-12
d an=4n-8

Answer 1

To find the nth term \( a_n \) of the sequence based on the given terms:

- \( a_3 = 0 \)
- \( a_4 = 4 \)
- \( a_5 = 8 \)
- \( a_6 = 12 \)
- \( a_7 = 16 \)

We can observe that this sequence is linear and increases by 4 for each increase in \( n \):

1. From \( a_3 \) to \( a_4 \): \( 0 \) to \( 4 \) (increase of 4)
2. From \( a_4 \) to \( a_5 \): \( 4 \) to \( 8 \) (increase of 4)
3. From \( a_5 \) to \( a_6 \): \( 8 \) to \( 12 \) (increase of 4)
4. From \( a_6 \) to \( a_7 \): \( 12 \) to \( 16 \) (increase of 4)

This indicates that the sequence can be represented by the equation:
\[
a_n = 4(n - 3)
\]
This can be simplified:
\[
a_n = 4n - 12
\]

Now we can look at the provided options to match our derived formula:

- a) \( a_n = -4n - 6 \)
- b) \( a_n = -4n - 12 \)
- c) \( a_n = 4n - 12 \) (This matches!)
- d) \( a_n = 4n - 8 \)

Thus, the correct equation that describes the nth term of the sequence is:
**c) \( a_n = 4n - 12 \)**.