Question 5 (3 marks)
The sum of the first two terms of an infinite geometric series is 45 . The third term in the
3 series is 12 .
Show that there are two possible series and find the first term and common ratio for each case.

Step 1 ：Let the first term be \(a\) and the common ratio be \(r\).

Step 2 ：Given, the sum of the first two terms is 45, so \(a + ar = 45\).

Step 3 ：Given, the third term is 12, so \(ar^2 = 12\).

Step 4 ：Solve the equations for \(a\) and \(r\):

Step 5 ：\(a(1 - r^2) = 45(1 - r)\)

Step 6 ：\(a = \frac{45(1 - r)}{1 - r^2}\)

Step 7 ：Substitute this value of \(a\) in the equation \(ar^2 = 12\):

Step 8 ：\(\frac{45(1 - r)}{1 - r^2}r^2 = 12\)

Step 9 ：Solve for \(r\) to get two possible values: \(r = \frac{2}{3}\) and \(r = -\frac{2}{5}\).

Step 10 ：Substitute these values of \(r\) in the equation for \(a\) to get the corresponding values of \(a\):

Step 11 ：For \(r = \frac{2}{3}\), \(a = 27\).

Step 12 ：For \(r = -\frac{2}{5}\), \(a = 75\).

Step 13 ：\(\boxed{\text{Final Answer: There are two possible series. The first series has a first term of 27 and a common ratio of }\frac{2}{3}\text{. The second series has a first term of 75 and a common ratio of }-\frac{2}{5}\text{.}}\)