Find the Missing Term in Powers Sequence: 729, _, 81, 27, 9, 3

To solve this problem, we'll analyze the sequence:

The sequence is given as $729, \_, 81, 27, 9, 3$.

• Step 1: Calculate the common ratio between consecutive terms.
• Step 2: Calculate the missing term using the calculated common ratio.

We notice that the known terms in the sequence become smaller, suggesting this is a decreasing series.

Step 1: Identify the common ratio

Check the ratio between consecutive known terms near the missing term:

Calculate the ratio between $81$ and $27$:

$r = \frac{27}{81} = \frac{1}{3}$

Calculate the ratio between $27$ and $9$:

$r = \frac{9}{27} = \frac{1}{3}$

Calculate the ratio between $9$ and $3$:

$r = \frac{3}{9} = \frac{1}{3}$

Each time, the ratio is $\frac{1}{3}$, confirming that it is indeed constant.

Step 2: Calculate the missing term

Now that we know the common ratio $r = \frac{1}{3}$, we calculate the missing term between $729$ and $81$.

$\frac{\_}{729} = \frac{1}{3}$

Multiplying both sides by $729$, the missing term is:

$\_ = 729 \times \frac{1}{3} = 243$

Therefore, the number written on the T-shirt of the fifth student is $243$.