9709 P1 - Jun 2002 - Q4
814
A progression has a first term of 12 and a fifth term of 18.
(i) Find the sum of the first 25 terms if the progression is arithmetic.
(ii) Find the 13th term if the progression is geometric.
Solution
(i) For the arithmetic progression, the first term is \(a = 12\) and the fifth term is \(a + 4d = 18\). Solving for \(d\), we have:
\(a + 4d = 18\)
\(12 + 4d = 18\)
\(4d = 6\)
\(d = 1.5\)
The sum of the first 25 terms \(S_{25}\) is given by:
\(S_{25} = \frac{25}{2} (2a + (25-1)d)\)
\(S_{25} = \frac{25}{2} (24 + 24 \times 1.5)\)
\(S_{25} = 750\)
(ii) For the geometric progression, the first term is \(a = 12\) and the fifth term is \(ar^4 = 18\). Solving for \(r\), we have:
\(ar^4 = 18\)
\(12r^4 = 18\)
\(r^4 = 1.5\)
The 13th term \(ar^{12}\) is given by:
\(ar^{12} = 12 \times (1.5)^3\)
\(ar^{12} = 40.5 \text{ or } 40.6\)
