A geometric series is a + ar + ar² + .. - Edexcel - A-Level Maths Pure - Question 1 - 2012 - Paper 3

Let the common ratio be denoted as r. We know that:

$r = \frac{T_5}{T_3} = \frac{1.944}{5.4}$

Calculating this gives:

$r = \frac{1.944}{5.4} \approx 0.36 \, (or \, 0.6)$

Using the third term:

$T_3 = ar^2 = 5.4$

Substituting the value of r:

$a(0.6)^2 = 5.4$

This simplifies to:

$a \cdot 0.36 = 5.4$

Thus we find:

$a = \frac{5.4}{0.36} = 15$

The sum to infinity for a geometric series is given by:

$S_{\infty} = \frac{a}{1 - r}$

Substituting the known values:

$S_{\infty} = \frac{15}{1 - 0.6} = \frac{15}{0.4} = 37.5$