Standard +0.3 This is a straightforward application of geometric series formulas with two equations in two unknowns. Part (a) requires setting up S∞ = a/(1-r) = 10 and recognizing that squaring terms gives a new GP with first term a² and ratio r², leading to a²/(1-r²) = 100/9. Solving these simultaneously is algebraically routine. Part (b) is immediate once (a) is solved, applying the same principle to cubes. While it requires understanding how squaring/cubing affects GP parameters, this is a standard textbook exercise with no novel insight required.
9. The sum to infinity of the geometric series
$$a + a r + a r ^ { 2 } + \ldots$$
is 10 .
The sum to infinity of the series formed by the squares of the terms is 100/9.
a) Show that \(r = 4 / 5\) and find \(a\).
b) Find the sum to infinity of the series formed by the cubes of the terms.
\section*{(Total for Question 9 is 5 marks)}
9. The sum to infinity of the geometric series
$$a + a r + a r ^ { 2 } + \ldots$$
is 10 .\\
The sum to infinity of the series formed by the squares of the terms is 100/9.\\
a) Show that $r = 4 / 5$ and find $a$.\\
b) Find the sum to infinity of the series formed by the cubes of the terms.
\section*{(Total for Question 9 is 5 marks)}
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q9 [5]}}