20N.2.SL.TZ0.S_6

pestleMathematicsAASLPaper 220N· sl-1-8-sum-of-infinite-geo-sequencesource ↗

An infinite geometric series has first term $u_{1} = a$ and second term $u_{2} = \frac{1}{4} a^{2} - 3 a$ , where $a > 0$ .

Find the common ratio in terms of $a$ .

[2]

Find the values of $a$ for which the sum to infinity of the series exists.

[3]

Find the value of $a$ when $S_{\infty} = 76$ .

[3]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

evidence of dividing terms (in any order) (M1)

eg $\frac{u_{1}}{u_{2}}, \frac{\frac{1}{4} a^{2} - 3 a}{a}$

$r = \frac{1}{4} a - 3$ A1 N2

[2 marks]

recognizing $|r| < 1$ (must be in terms of $a$ ) (M1)

eg $|\frac{1}{4} a - 3| < 1, - 1 \leq \frac{1}{4} a - 3 \leq 1, - 4 < a - 12 < 4$

$8 < a < 16$ A2 N3

[3 marks]

correct equation (A1)

eg $\frac{a}{1 - (\frac{1}{4} a - 3)} = 76, a = 76 (4 - \frac{1}{4} a)$

$a = \frac{76}{5} (= 15.2)$ (exact) A2 N3

[3 marks]

Examiners’ report

[N/A]