The second and third terms of a geometric series are 192 and 144 respectively - Edexcel - A-Level Maths Pure - Question 8 - 2011 - Paper 2

We can find the first term by rearranging the equation for the second term:

$ar = 192$

Substituting the value of r we found:

$a(0.75) = 192$

Solving for a, we have:

$a = \frac{192}{0.75} = 256$

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

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

Where a is the first term and r is the common ratio. Substituting the values we found:

$S_\infty = \frac{256}{1 - 0.75} = \frac{256}{0.25} = 1024$

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

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

From previous calculations:

$S_n = 256 \frac{1 - (0.75)^n}{1 - 0.75} = 256 \frac{1 - (0.75)^n}{0.25} = 1024(1 - (0.75)^n)$

To find the smallest n such that:

$1024(1 - (0.75)^n) > 1000$

This simplifies to:

$1 - (0.75)^n > \frac{1000}{1024}$

Which leads to:

$(0.75)^n < 0.0234375$

Taking logarithms on both sides:

$n \log(0.75) < \log(0.0234375)$

This implies:

$n > \frac{\log(0.0234375)}{\log(0.75)}$

Calculating:

$n \approx 14$

Thus, the smallest value of n is 14.