The sum of the first three terms of a geometric sequence is $62.755$, and the sum of the infinite sequence is $440$. Find the common ratio.

correct substitution into sum of a geometric sequence     A1

eg   $62.755 = {u_1}\left( {\frac{{1 - {r^3}}}{{1 - r}}} \right)$ , ${u_1} + {u_1}r + {u_1}{r^2} = 62.755$

correct substitution into sum to infinity     A1

eg   $\frac{{{u_1}}}{{1 - r}} = 440$

attempt to eliminate one variable     (M1)

eg substituting ${u_1} = 440(1 - r)$

correct equation in one variable     (A1)

eg   $62.755 = 440(1 - r)\left( {\frac{{1 - {r^3}}}{{1 - r}}} \right)$ , $440(1 - r)(1 + r + {r^2}) = 62.755$

evidence of attempting to solve the equation in a single variable     (M1)

eg sketch, setting equation equal to zero, $62.755 = 440(1 - {r^3})$

$r =0.95 = \frac{{19}}{{20}}$     A1     N4

[6 marks]

Many candidates were able to successfully obtain two equations in two variables, but far fewer were able to correctly solve for the value of $r$. Some candidates had misread errors for either $440$ or $62.755$, with some candidates taking the French and Spanish exams mistaking the decimal comma for a thousands comma. Many candidates who attempted to solve algebraically did not cancel the $1 - r$ from both sides and ended up with a 4^th degree equation that they could not solve. Some of these candidates obtained the extraneous answer of $r - 1$ as well. Some candidates used a minimum of algebra to eliminate the first term and then quickly solved the resulting equation on their GDC.