\(a=4\), \(r=12/4=3\).

\[ S_{10}=\frac{a(r^{10}-1)}{r-1} =\frac{4(3^{10}-1)}{3-1} =\frac{4(59049-1)}{2} =118096. \]

\(u_3=ar^2=5\), \(u_4=ar^3=-20\).

\(r=u_4/u_3=-20/5=-4\).

\(a=5/r^2=5/16\).

\(u_9=ar^8=(5/16)(-4)^8=(5/16)\cdot 65536=20480\).

\(u_4=ar^3=200\), \(u_6=ar^5=800\).

Divide: \(r^2=800/200=4\Rightarrow r=2\) or \(r=-2\).

If \(r=2\): \(a=200/2^3=25\).

If \(r=-2\): \(a=200/(-2)^3=-25\).

\(a=1.1\), \(r=1.21/1.1=1.1\).

\[ u_{13}=ar^{12}=1.1(1.1)^{12}=1.1^{13}\approx 3.45227. \]

3 s.f. \(=3.45\).

\(a=\frac12\), \(r=-\frac12\), \(n=11\).

\[ S_{11}=\frac{a(1-r^{11})}{1-r} =\frac{\frac12\bigl(1-(-\frac12)^{11}\bigr)}{1+\frac12} =\frac{\frac12\left(1+\frac1{2048}\right)}{\frac32} =\frac{683}{2048}. \]

\(u_5=ar^4=0.0016\), with \(a=1\).

So \(r^4=0.0016=(0.2)^4\Rightarrow r=0.2\) or \(r=-0.2\).

For \(r=0.2\):

\[ S_8=\frac{1-0.2^8}{1-0.2} \approx \frac{0.99999744}{0.8} \approx 1.25. \]

For \(r=-0.2\):

\[ S_8=\frac{1-(-0.2)^8}{1+0.2} \approx \frac{0.99999744}{1.2} \approx 0.833. \]

\(r=\frac25\), \(S_4=2\frac{13}{25}=\frac{63}{25}\).

\[ S_4=\frac{a(1-r^4)}{1-r} \Rightarrow \frac{63}{25}=a\cdot\frac{1-(\frac25)^4}{1-\frac25}. \]

\(r^4=\frac{16}{625}\), so \(1-r^4=\frac{609}{625}\), and \(1-r=\frac35\).

\[ \frac{63}{25}=a\cdot\frac{609}{625}\cdot\frac{5}{3} \Rightarrow a=\frac{45}{29}. \]

\(u_4=ar^3\), \(u_6=ar^5\). Given \(ar^3=9ar^5\Rightarrow r^2=\frac19\).

Since \(r>0\), \(r=\frac13\).

\(u_5=ar^4=3\Rightarrow a(\frac13)^4=3\Rightarrow a=243\).

\[ S_6=\frac{a(1-r^6)}{1-r} =\frac{243\left(1-(\frac13)^6\right)}{1-\frac13} =364. \]

\(u_1=(-\frac13)^1=-\frac13\Rightarrow a=-\frac13\).

\(r=u_2/u_1=(-\frac13)^2/(-\frac13)= -\frac13\).

\[ S_9=\frac{a(1-r^9)}{1-r},\quad r=-2. \]

\(r^9=(-2)^9=-512\Rightarrow 1-r^9=513\), and \(1-r=3\).

\[ 684=a\cdot\frac{513}{3}=171a\Rightarrow a=4. \]

\[ u_8=ar^7=4(-2)^7=4(-128)=-512. \]