Problem Statement
An infinite geometric series has first term $u_1 = 18$ and second term $u_2 = 12$.
**(a)** [1 marks] Find the common ratio $r$.
**(b)** [2 marks] Find the exact value of the tenth term $u_{10}$.
**(c)** [2 marks] Find the sum to infinity, $S_{\infty}$.
Verified Solution & Marking Scheme
Part (a) [A1]
$r = \frac{u_2}{u_1} = \frac{12}{18} = \frac{2}{3}$
Part (b) [M1]
Using $u_n = u_1 r^{n-1} = 18 \left(\frac{2}{3}\right)^9$
Part (b) [A1]
$u_{10} = \frac{1024}{2187}$ (or $18 \times \left(\frac{2}{3}\right)^9$)
Part (b) [Note]
Accept exact fraction $\frac{1024}{2187}$ or unsimplified form $18\left(\frac{2}{3}\right)^9$.
Part (c) [M1]
Using $S_{\infty} = \frac{u_1}{1 - r} = \frac{18}{1 - 2/3}$
Part (c) [A1]
$S_{\infty} = \frac{18}{1/3} = 54$
Part (c) [Note]
Award M0 if candidate attempts arithmetic series formula $S_n = \frac{n}{2}(2a + (n-1)d)$.