Question

the sequence $a_n = 500 + 75(n - 1)$ represents the production of shirts in a factory over a period of $n$ weeks. which table corresponds to this sequence? (four tables are shown with columns $n$ and $a_n$, rows for $n=1$ to $n=5$ with different values.)

Explanation:

The sequence is given by \( a_n = 500 + 75(n - 1) \). Let's simplify this formula first.

Step 1: Simplify the formula

Simplify \( a_n = 500 + 75(n - 1) \):

$$
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Step 2: Check for \( n = 1 \)

Substitute \( n = 1 \) into the simplified formula:

$$ a_1 = 425 + 75(1) = 425 + 75 = 500 $$

Wait, that doesn't match the first table. Wait, maybe I misread the original formula. Wait, the original problem says "The sequence \( a_n = 500 + 75(n - 1) \)". Wait, let's recalculate:

Wait, \( 500 + 75(1 - 1) = 500 + 0 = 500 \) for \( n = 1 \). Then \( n = 2 \): \( 500 + 75(2 - 1) = 500 + 75 = 575 \). \( n = 3 \): \( 500 + 75(3 - 1) = 500 + 150 = 650 \). \( n = 4 \): \( 500 + 75(4 - 1) = 500 + 225 = 725 \). \( n = 5 \): \( 500 + 75(5 - 1) = 500 + 300 = 800 \). Wait, but the third table (from left) has:

First table (leftmost): \( n=1 \): 600, \( n=2 \): 425... No. Second table: \( n=1 \): 575, \( n=2 \): 500... No. Third table: \( n=1 \): 500, \( n=2 \): 575, \( n=3 \): 650, \( n=4 \): 725, \( n=5 \): 800. Let's check with the formula:

For \( n=1 \): \( 500 + 75(0) = 500 \) ✔️

\( n=2 \): \( 500 + 75(1) = 575 \) ✔️

\( n=3 \): \( 500 + 75(2) = 650 \) ✔️

\( n=4 \): \( 500 + 75(3) = 725 \) ✔️

\( n=5 \): \( 500 + 75(4) = 800 \) ✔️

So the third table (counting from left, the third one) matches the sequence \( a_n = 500 + 75(n - 1) \).

Wait, let's check each table:

First table (leftmost):

\( n=1 \): 600, \( n=2 \): 425... Not matching.

Second table:

\( n=1 \): 575, \( n=2 \): 500... \( 500 + 75(1-1)=500 \), so no.

Third table:

\( n=1 \): 500, \( n=2 \): 575, \( n=3 \): 650, \( n=4 \): 725, \( n=5 \): 800. Which matches \( a_n = 500 + 75(n - 1) \).

Fourth table (rightmost):

\( n=1 \): 600, \( n=2 \): 425... No.

So the correct table is the third one (the one with \( n=1 \): 500, \( n=2 \): 575, \( n=3 \): 650, \( n=4 \): 725, \( n=5 \): 800).

Answer:

The third table (the one with \( n = 1 \) having \( a_n = 500 \), \( n = 2 \) having \( a_n = 575 \), \( n = 3 \) having \( a_n = 650 \), \( n = 4 \) having \( a_n = 725 \), \( n = 5 \) having \( a_n = 800 \))