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]