Question

Consider the sequence 6,3,0,-3,-6,-9….
What is the function that represents the sequence for the term numbers 1,2,3,4,5, and 6

Answer 1

The options are
f(1)=0; f(n+1)=3+f(n)
f(1)=6; f(n)=f(n-1)-3
f(1)=6; f(n+1)=1/3f(n)
f(1)=-9; f(n)=3f(n-1)

Answer 2

The given sequence is 6, 3, 0, -3, -6, -9. To find a function that represents the nth term of this sequence, we first observe the pattern of the terms:

- The first term (n=1) is 6.
- The second term (n=2) is 3.
- The third term (n=3) is 0.
- The fourth term (n=4) is -3.
- The fifth term (n=5) is -6.
- The sixth term (n=6) is -9.

We can see that the sequence decreases by 3 for each subsequent term. This suggests that the sequence is linear, and we can express it in the form:
\[
a_n = a_1 + (n-1) \cdot d
\]
where \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

Here:
- \( a_1 = 6 \)
- \( d = -3 \)

Substituting these values into the formula gives us:
\[
a_n = 6 + (n-1)(-3) = 6 - 3(n-1)
\]
Simplifying this further:
\[
a_n = 6 - 3n + 3 = 9 - 3n
\]

Thus, the function that represents the nth term of the sequence is:
\[
a_n = 9 - 3n
\]

This function matches the given terms for \( n = 1, 2, 3, 4, 5, \) and \( 6 \).

Answer 3

Let's analyze the given options based on the sequence we discussed earlier: 6, 3, 0, -3, -6, -9.

1. **Option 1**: \( f(1) = 0; \, f(n+1) = 3 + f(n) \)

- If \( f(1) = 0 \), then \( f(2) = 3 + 0 = 3 \)
- \( f(3) = 3 + 3 = 6 \) (incorrect)
- This option does not match the sequence.

2. **Option 2**: \( f(1) = 6; \, f(n) = f(n-1) - 3 \)

- If \( f(1) = 6 \), then:
- \( f(2) = 6 - 3 = 3 \)
- \( f(3) = 3 - 3 = 0 \)
- \( f(4) = 0 - 3 = -3 \)
- \( f(5) = -3 - 3 = -6 \)
- \( f(6) = -6 - 3 = -9 \)
- This option matches the sequence perfectly.

3. **Option 3**: \( f(1) = 6; \, f(n+1) = \frac{1}{3}f(n) \)

- If \( f(1) = 6 \):
- \( f(2) = \frac{1}{3} \cdot 6 = 2 \)
- \( f(3) = \frac{1}{3} \cdot 2 = \frac{2}{3} \) (incorrect)
- This does not match the sequence.

4. **Option 4**: \( f(1) = -9; \, f(n) = 3f(n-1) \)

- If \( f(1) = -9 \):
- \( f(2) = 3 \cdot (-9) = -27 \) (incorrect)
- This does not match the sequence.

Based on the analysis, the only option that correctly represents the sequence is:

**Option 2**: \( f(1) = 6; \, f(n) = f(n-1) - 3 \)