The first three terms of a geometric series are 18, 12 and p respectively, where p is a constant - Edexcel - A-Level Maths Pure - Question 3 - 2013 - Paper 4

In a geometric series, the relationship between the terms can be represented as:

$p = a_2 \cdot r = 12 \cdot \frac{2}{3} = 8$

Thus, the value of p is 8.

The formula for the sum of the first n terms of a geometric series is:

$S_n = \frac{a(1 - r^n)}{1 - r}$

Where:

• a = first term = 18
• r = common ratio = $\frac{2}{3}$
• n = number of terms = 15

Now, substituting the values into the formula:

$S_{15} = \frac{18(1 - (\frac{2}{3})^{15})}{1 - \frac{2}{3}}$

Calculating:

• $\frac{1}{1 - \frac{2}{3}} = 3$
• The term $\left(\frac{2}{3}\right)^{15}$ can be calculated to find its value.

Thus, $S_{15} = 18 \cdot 3 \cdot \left(1 - (\frac{2}{3})^{15}\right)\approx 53.877$

Therefore, the sum of the first 15 terms of the series is approximately 53.877.