Question

In a sequence of numbers, `a_{3}=0,\ a_{4}=4,\ a_{5}=8,\ a_{6}=12,\ a_{7}=16.`

Based on this information, which equation can be used to find the `n^{th}` term in the sequence, `a_{n}`?
a. a_{n}=-4n-6
b. a_{n}=-4n-12
c. a_{n}=4n-12
d. a_{n}=4n-8

Answer 1

To find the equation for the \( n^{th} \) term in the sequence given the specific terms, we can first observe the terms provided:

- \( a_{3} = 0 \)
- \( a_{4} = 4 \)
- \( a_{5} = 8 \)
- \( a_{6} = 12 \)
- \( a_{7} = 16 \)

We can see that starting from \( a_4 \), the sequence increases by 4 for each subsequent term:

- \( a_{4} = 4 \)
- \( a_{5} = 8 \) (which is \( 4 + 4 \))
- \( a_{6} = 12 \) (which is \( 8 + 4 \))
- \( a_{7} = 16 \) (which is \( 12 + 4 \))

This indicates a linear pattern suggesting that the sequence can be represented by the equation of a line, possibly of the form \( a_n = 4(n - k) \) for some \( k \).

Now, to derive a specific equation, we look for a way to express \( a_n \) based on the given values. Specifically, to derive an equation for \( n \):
- From \( a_{3} = 0 \), we find that the sequence starts at \( n = 3 \) and thus can be framed as a function that relates to \( n \) starting from 3.

Since we recognize that the terms increment by 4 starting from \( n = 4 \), we can look for a pattern:

Calculating the direct relationship:
- When \( n = 4 \), \( a_4 = 4 \)
- When \( n = 5 \), \( a_5 = 8 \)
- When \( n = 6 \), \( a_6 = 12 \)
- When \( n = 7 \), \( a_7 = 16 \)

To derive the formula starting from \( n = 3 \):

From \( n = 3 \) to \( n = 4 \):
\( a_n = 4(n - 1) - 4 \) or rearranging \( a_n = 4n - 12 \).

Now, checking the candidates you've given:
- When substituting \( n = 4 \):
- Option c: \( a_n = 4 \cdot 4 - 12 = 16 - 12 = 4 \) (correct)
- For \( n = 3 \):
- \( a_n = 4 \cdot 3 - 12 = 12 - 12 = 0 \) (correct)

Hence, the equation that can be used to find the \( n^{th} \) term in the sequence, \( a_n \), is:
**c. \( a_{n} = 4n - 12 \)**