Find the common ratio in terms of $a$.

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

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

* 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       $\frac{u_1}{u_2}, \frac{{\displaystyle \frac{1}{4}{a}^2-3a}}{a}$

$r=\frac{1}{4}a-3$      A1   N2

[2 marks]

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

eg       $\left|\frac{1}{4}a-3\right|<1, -1\le \frac{1}{4}a-3\le 1, -4<a-12<4$

$8<a<16$      A2   N3

[3 marks]

correct equation     (A1)

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

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

[3 marks]