If a 3'' by 3'' square is added at each successive stage, what will be the area of the rectangle at Stage 6, in square inches?

[asy]size(250); real textsize = 10pt;
draw(unitsquare);
draw(shift(1.5*right)*unitsquare);
draw(shift(2.5*right)*unitsquare);
draw(shift(4*right)*unitsquare);
draw(shift(5*right)*unitsquare);
draw(shift(6*right)*unitsquare);
label("Stage 1",(.5,0),S,fontsize(textsize));
label("Stage 2",(2.5,0),S,fontsize(textsize));
label("Stage 3",(5.5,0),S,fontsize(textsize));[/asy]
The number of squares in the rectangles is an arithmetic sequence with first term 1 and common difference 1. Thus, at Stage 6, there will be 6 squares. Since each square has an area of $3 \cdot 3 = 9$ square inches, the total area of the rectangle at Stage 6 is $6 \cdot 9 = \boxed{54}$ square inches.
54
Level 2
Algebra
Problem: If a 3'' by 3'' square is added at each successive stage, what will be the area of the rectangle at Stage 6, in square inches?

[asy]size(250); real textsize = 10pt;
draw(unitsquare);
draw(shift(1.5*right)*unitsquare);
draw(shift(2.5*right)*unitsquare);
draw(shift(4*right)*unitsquare);
draw(shift(5*right)*unitsquare);
draw(shift(6*right)*unitsquare);
label("Stage 1",(.5,0),S,fontsize(textsize));
label("Stage 2",(2.5,0),S,fontsize(textsize));
label("Stage 3",(5.5,0),S,fontsize(textsize));[/asy]
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the area of the rectangle at Stage 6, we first need to understand the pattern of area growth as squares are added. Each stage adds a 3'' by 3'' square to the existing shape. 

At Stage 1, the area is simply the area of one square, which is 3 inches * 3 inches = 9 square inches. 

At Stage 2, we add another square, so the total area becomes 9 square inches + 9 square inches = 18 square inches. 

Continuing this pattern, at Stage 3, we add yet another square, resulting in an area of 18 square inches + 9 square inches = 27 square inches. 

This process continues, with each stage adding an additional 9 square inches to the total area. 

To find the area at Stage 6, we can observe that the area increases by 9 square inches for each stage. Thus, we can express the area at Stage n as:

Area at Stage n = 9 square inches * n.

For Stage 6, we substitute n with 6:

Area at Stage 6 = 9 square inches * 6 = 54 square inches.

Therefore, the final answer is 54.<|END_OF_LATENT|> The final answer is $54$.