Question

write a recursive formula and an explicit formula for the arithmetic sequence shown in the graph. a recursive formula is a₁ = □, aₙ = □. (simplify your answers.)

Explanation:

Step1: Identify the first term

From the graph, when \( n = 1 \), the value of \( a_1 \) is 10 (looking at the first point, likely at \( x = 1 \), \( y = 10 \)? Wait, wait, the first point: let's check the coordinates. Wait, the first point (n=1) seems to be (1,10)? Wait no, the first orange dot: let's see the grid. Wait, maybe I misread. Wait, the first point (n=1) is at y=10? Wait no, the first dot: looking at the graph, the first dot (when x=1) is at y=10? Wait, no, the first dot (n=1) is at y=10? Wait, maybe the first term \( a_1 = 10 \)? Wait, no, let's check the sequence. Wait, the points: when n=1, y=10; n=2, y=8; n=3, y=6; n=4, y=4? Wait, no, the first dot: let's see the graph. Wait, the first dot (x=1) is at y=10? Wait, no, the first dot (x=1) is at y=10? Wait, maybe the first term \( a_1 = 10 \), and the common difference \( d = -2 \), because each subsequent term decreases by 2. Wait, let's confirm. If n=1: 10, n=2: 8, n=3: 6, n=4: 4. So the common difference \( d = 8 - 10 = -2 \). So the recursive formula for an arithmetic sequence is \( a_1 = \text{first term} \), and \( a_n = a_{n - 1} + d \). So here, \( a_1 = 10 \), and \( a_n = a_{n - 1} - 2 \) (since \( d = -2 \)). Wait, but let's check the graph again. Wait, maybe the first point is (1,10)? Wait, the first dot: when x=1, y=10? Then x=2, y=8; x=3, y=6; x=4, y=4. So yes, that's an arithmetic sequence with first term 10 and common difference -2. So the recursive formula is \( a_1 = 10 \), \( a_n = a_{n - 1} - 2 \) for \( n \geq 2 \).

Step2: Confirm the recursive formula

For an arithmetic sequence, the recursive formula is \( a_1 = \text{first term} \), and \( a_n = a_{n - 1} + d \), where \( d \) is the common difference. Here, \( a_1 = 10 \), \( d = -2 \), so \( a_n = a_{n - 1} - 2 \).

Answer:

\( a_1 = 10 \), \( a_n = a_{n - 1} - 2 \) (for \( n \geq 2 \))