F.1 Mathematics

Question 1
A cyclist, whose average speed is 10 km/h sets out to ride from X to Y. At the same time, another cyclist, whose average speed is 8 km/h, sets out to ride from Y to X. If they meet 2 km from half-way, find the distance between X and Y.

Solution 1
Let d km be the distance between X and Y.
Let t h be the time before the two cyclists meet.

Recall that Distance = Speed x Time

{ 10 t + 8 t = d ...[1]
{ 10 t - d/2 = 2 ...[2]

From [1], 18t = d, t = d/18, put into [2].

10 (d/18) - d/2 = 2

5d/9 - d/2 = 2

Multiply 18 to both sides.

10d - 9d = 36

d = 36

The distance between X and Y is 36 km.


Question 2
Consider the following sequence.
2, 9, 28, 65, 126
Guess the general term of the sequence, and check by the method of substitution.

Solution 2
Observe:
2 = 1 + 1 = 1³ + 1
9 = 8 + 1 = 2³ + 1
28 = 27 + 1 = 3³ + 1
65 = 64 + 1 = 4³ + 1
126 = 125 + 1 = 5³ + 1

Guess the general term of the sequence is T(n) = n³ + 1.

Check that
T(1) = 1³ + 1 = 1 + 1 = 2
T(2) = 2³ + 1 = 8 + 1 = 9
T(3) = 3³ + 1 = 27 + 1 = 28
T(4) = 4³ + 1 = 64 + 1 = 65
T(5) = 5³ + 1 = 125 + 1 = 126

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