A car has six forward gears - Edexcel - A-Level Maths Pure - Question 7 - 2020 - Paper 1

To find the fastest speed of the car in the 5th gear, we use the formula for the nth term of a geometric sequence:

$a_n = a r^{(n - 1)}$

where:

• $a$ is the first term (the speed in 1st gear, which is 28 km h⁻¹)
• $r$ is the common ratio
• $n$ is the term number (in this case, 5 for the 5th gear)

We know:

• $a_6 = 115$ km h⁻¹

Using the formula for the 6th term:

115 = 28 r^5 $$ Now we solve for $r^5$: $$ r^5 = \frac{115}{28} \ = 4.10714 \

Taking the 5th root gives:

$$$$

Now substituting back into the formula for the 5th term:

$a_5 = 28 \times r^{(5 - 1)} \ = 28 \times (1.3265)^{4} \ = 28 \times 3.228 \approx 90.4 \ { ext{km h}}^{-1}$

Thus, the fastest speed of the car in 5th gear is approximately 86.7 km h⁻¹.