Find the common ratio.

Solve \(\sum\limits_{k = 1}^\infty  {{2^{5 - k}}\ln x = 64} \).

correct use \(\log {x^n} = n\log x\)     A1

eg\(\,\,\,\,\,\)\(16\ln x\)

valid approach to find \(r\)     (M1)

eg\(\,\,\,\,\,\)\(\frac{{{u_{n + 1}}}}{{{u_n}}},{\text{ }}\frac{{\ln {x^8}}}{{\ln {x^{16}}}},{\text{ }}\frac{{4\ln x}}{{8\ln x}},{\text{ }}\ln {x^4} = \ln {x^{16}} \times {r^2}\)

\(r = \frac{1}{2}\)     A1     N2

[3 marks]

recognizing a sum (finite or infinite)     (M1)

eg\(\,\,\,\,\,\)\({2^4}\ln x + {2^3}\ln x,{\text{ }}\frac{a}{{1 - r}},{\text{ }}{S_\infty },{\text{ }}16\ln x +  \ldots \)

valid approach (seen anywhere)     (M1)

eg\(\,\,\,\,\,\)recognizing GP is the same as part (a), using their \(r\) value from part (a), \(r = \frac{1}{2}\)

correct substitution into infinite sum (only if \(\left| r \right|\) is a constant and less than 1)     A1

eg\(\,\,\,\,\,\)\(\frac{{{2^4}\ln x}}{{1 - \frac{1}{2}}},{\text{ }}\frac{{\ln {x^{16}}}}{{\frac{1}{2}}},{\text{ }}32\ln x\)

correct working     (A1)

eg\(\,\,\,\,\,\)\(\ln x = 2\)

\(x = {{\text{e}}^2}\)     A1     N3

[5 marks]