Standard +0.8 This is a multi-step geometric progression problem requiring simultaneous equations (using ar^6 = 2ar^4 and the sum formula), solving a quadratic for r, then substituting back to find 'a' in surd form. It demands algebraic manipulation beyond routine GP questions and requires careful handling of the surd simplification, making it moderately challenging but within standard Further Maths scope.
The seventh term of a geometric progression is equal to twice the fifth term. The sum of the first seven terms is 254 and the terms are all positive. Find the first term, showing that it can be written in the form \(p + q\sqrt{r}\) where \(p\), \(q\) and \(r\) are integers. [6]
The seventh term of a geometric progression is equal to twice the fifth term. The sum of the first seven terms is 254 and the terms are all positive. Find the first term, showing that it can be written in the form $p + q\sqrt{r}$ where $p$, $q$ and $r$ are integers. [6]
\hfill \mbox{\textit{SPS SPS FM 2023 Q7 [6]}}