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Date May 2016 Marks available 6 Reference code 16M.2.hl.TZ2.4
Level HL only Paper 2 Time zone TZ2
Command term Find Question number 4 Adapted from N/A

Question

The sum of the second and third terms of a geometric sequence is 96.

The sum to infinity of this sequence is 500.

Find the possible values for the common ratio, \(r\).

Markscheme

\(ar + a{r^2} = 96\)    A1

Note:     Award A1 for any valid equation involving \(a\) and \(r\), eg, \(\frac{{a(1 - {r^3})}}{{1 - r}} - a = 96\).

\(\frac{a}{{1 - r}} = 500\)    A1

EITHER

attempting to eliminate \(a\) to obtain \(500r(1 - {r^2}) = 96\) (or equivalent in unsimplified form)     (M1)

OR

attempting to obtain \(a = \frac{{96}}{{r + {r^2}}}\) and \(a = 500(1 - r)\)     (M1)

THEN

attempting to solve for \(r\)     (M1)

\(r = 0.2{\text{ }}\left( { = \frac{1}{5}} \right)\) or \(r = 0.885{\text{ }}\left( { = \frac{{\sqrt {97}  - 1}}{{10}}} \right)\)     A1A1

[6 marks]

Examiners report

Reasonably well done. Quite a number of candidates included a solution outside \( - 1 < r < 1\).

Syllabus sections

Topic 1 - Core: Algebra » 1.1 » Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series.
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