Extending a rectangle

A rectangle will be repeatedly enlarged such that the side which is at the given moment the shorter we extend by 3 cm and the longer side only by 1 cm.
After the third extension, a rectangle with dimensions 11 cm and 12 cm is formed.
1. Determine the dimensions of the original rectangle.
2. Determine the dimensions of the rectangle after the fifth extension.
3. Determine the dimensions of the rectangle after the one hundred and fifth extension.

(The result of the 1st and 2nd questions is easy to gradually derive, but what principle to apply in order to get to any n-th extension)

Final Answer:

a₁ =  6 cm
b₁ =  5 cm
a₆ =  15 cm
b₆ =  16 cm
a₁₀₆ =  215 cm
b₁₀₆ =  216 cm

Step-by-step explanation:

$$\begin{gathered} a_4=11\text{cm} \\ {b}_4=12\text{cm} \\ {a}_3=a_4-1=11-1=10\text{cm} \\ {b}_3=b_4-3=12-3=9\text{cm} \\ {a}_2=a_3-3=10-3=7\text{cm} \\ {b}_2=b_3-1=9-1=8\text{cm} \\ {a}_1=a_2-1=7-1=6\text{cm} \end{gathered}$$

$b_1=b_2-3=8-3=5\text{cm}$

$$\begin{gathered} a_5=a_4+3=11+3=14\text{cm} \\ {b}_5=b_4+1=12+1=13\text{cm} \\ {a}_6=a_5+1=14+1=15\text{cm} \end{gathered}$$

$b_6=b_5+3=13+3=16\text{cm}$

$$\begin{gathered} m=(3+1)/2=2\text{cm} \\ {a}_{106}=a_6+100\cdot m=15+100\cdot 2=215\text{cm} \end{gathered}$$

$b_{106}=b_6+100\cdot m=16+100\cdot 2=216\text{cm}$

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