Question

1. which equation represents the nth term of the sequence {2, -1, -4, -7, ...}? a. ( a_n = 2n - 5 ) b. ( a_n = 3n - 1 ) c. ( a_n = -3n - 10 ) d. ( a_n = -3n + 5 )

(top right: a. -6, b. -3, c. 2, d. 3 (circled d))
use for questions 29 and 30: darius filled up his drives, he uses 0.06 gallons of gas.

1. which formula represents the number of...

Explanation:

Step1: Identify the sequence type

The sequence is \(2, -1, -4, -7, \dots\). It's an arithmetic sequence with first term \(a_1 = 2\) and common difference \(d=-3\) (since \(-1 - 2=-3\), \(-4 - (-1)=-3\), etc.).

Step2: Use arithmetic sequence formula

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\). Substitute \(a_1 = 2\) and \(d=-3\):
\[

$$\begin{align*} a_n&=2+(n - 1)(-3)\\ &=2-3n + 3\\ &=-3n+5 \end{align*}$$

\]

Step3: Verify with options

Check each option:

• Option A: \(a_n = 2n-5\). For \(n = 1\), \(2(1)-5=-3

eq2\). Eliminate.

• Option B: \(a_n = 3n - 1\). For \(n = 1\), \(3(1)-1 = 2\), but for \(n = 2\), \(3(2)-1 = 5

eq-1\). Eliminate.

• Option C: \(a_n=-3n - 10\). For \(n = 1\), \(-3(1)-10=-13

eq2\). Eliminate.

• Option D: \(a_n=-3n + 5\). For \(n = 1\), \(-3(1)+5 = 2\); \(n = 2\), \(-3(2)+5=-1\); \(n = 3\), \(-3(3)+5=-4\); \(n = 4\), \(-3(4)+5=-7\). Matches the sequence.

Answer:

D. \(a_n=-3n + 5\)