Question

which sequence is generated by the function $f(n + 1) = f(n) - 2$ for $f(1) = 10$?

• $-2, 8, 18, 28, 38, \dots$
• $-10, -12, -14, -16, -18,$
• $10, 8, 6, 4, 2, \dots$
• $8, 18, 28, 38, 48, \dots$

Explanation:

Step1: Understand the recursive formula

The function is given by \( f(n + 1)=f(n)-2 \) with \( f(1) = 10 \). This means each term in the sequence is obtained by subtracting 2 from the previous term.

Step2: Find the first few terms

• For \( n = 1 \), we know \( f(1)=10 \).
• For \( n = 2 \), using the formula \( f(2)=f(1)-2 \). Substituting \( f(1) = 10 \), we get \( f(2)=10 - 2=8 \).
• For \( n = 3 \), \( f(3)=f(2)-2 \). Substituting \( f(2) = 8 \), we get \( f(3)=8 - 2 = 6 \).
• For \( n = 4 \), \( f(4)=f(3)-2 \). Substituting \( f(3)=6 \), we get \( f(4)=6 - 2=4 \).
• For \( n = 5 \), \( f(5)=f(4)-2 \). Substituting \( f(4) = 4 \), we get \( f(5)=4 - 2=2 \).

So the sequence starts with 10, 8, 6, 4, 2, ...

Answer:

10, 8, 6, 4, 2, ... (the third option: 10, 8, 6, 4, 2, ...)