The following table gives the values of \({x_n}\) and \({A_n}\), for \(1 \leqslant n \leqslant 3\). Copy and complete the table. (Do not write on this page.)

The process described above is repeated. Find \({A_6}\).

Consider an initial square of side length \(k {\text{ cm}}\). The process described above is repeated indefinitely. The total area of the shaded regions is \(k {\text{ c}}{{\text{m}}^2}\). Find the value of \(k\).

valid method for finding side length     (M1)

eg   \({8^2} + {8^2} = {c^2},{\text{ }}45 - 45 - 90{\text{ side ratios, }}8\sqrt 2 ,{\text{ }}\frac{1}{2}{s^2} = 16,{\text{ }}{x^2} + {x^2} = {8^2}\)

correct working for area     (A1)

eg   \(\frac{1}{2} \times 4 \times 4\)

     A1A1     N2N2

[4 marks]

METHOD 1

recognize geometric progression for \({A_n}\)     (R1)

eg   \({u_n} = {u_1}{r^{n - 1}}\)

\(r = \frac{1}{2}\)     (A1)

correct working     (A1)

eg   \(32{\left( {\frac{1}{2}} \right)^5};{\text{ 4, 2, 1, }}\frac{1}{2},{\text{ }}\frac{1}{4},{\text{ }} \ldots \)

\({A_6} = 1\)     A1     N3

METHOD 2

attempt to find \({x_6}\)     (M1)

eg   \(8{\left( {\frac{1}{{\sqrt 2 }}} \right)^5},{\text{ }}2\sqrt 2 ,{\text{ 2, }}\sqrt 2 ,{\text{ 1, }} \ldots \)

\({x_6} = \sqrt 2 \)     (A1)

correct working     (A1)

eg   \(\frac{1}{2}{\left( {\sqrt 2 } \right)^2}\)

\({A_6} = 1\)     A1     N3

[4 marks]

METHOD 1

recognize infinite geometric series     (R1)

eg   \({S_n} = \frac{a}{{1 - r}},{\text{ }}\left| r \right| < 1\)

area of first triangle in terms of \(k\)     (A1)

eg   \(\frac{1}{2}{\left( {\frac{k}{2}} \right)^2}\)

attempt to substitute into sum of infinite geometric series (must have \(k\))     (M1)

eg   \(\frac{{\frac{1}{2}{{\left( {\frac{k}{2}} \right)}^2}}}{{1 - \frac{1}{2}}},{\text{ }}\frac{k}{{1 - \frac{1}{2}}}\)

correct equation     A1

eg   \(\frac{{\frac{1}{2}{{\left( {\frac{k}{2}} \right)}^2}}}{{1 - \frac{1}{2}}} = k,{\text{ }}k = \frac{{\frac{{{k^2}}}{8}}}{{\frac{1}{2}}}\)

correct working     (A1)

eg   \({k^2} = 4k\)

valid attempt to solve their quadratic     (M1)

eg   \(k(k - 4),{\text{ }}k = 4{\text{ or }}k = 0\)

\(k = 4\)     A1     N2

METHOD 2

recognizing that there are four sets of infinitely shaded regions with equal area     R1

area of original square is \({k^2}\)     (A1)

so total shaded area is \(\frac{{{k^2}}}{4}\)     (A1)

correct equation \(\frac{{{k^2}}}{4} = k\)     A1

\({k^2} = 4k\)    (A1)

valid attempt to solve their quadratic     (M1)

eg   \(k(k - 4),{\text{ }}k = 4{\text{ or }}k = 0\)

\(k = 4\)     A1     N2

[7 marks]