TOPIC I. NUMBERS & ALGEBRA Subtopic: SEQUENCES

FACT: In a 60 kph speed zone, the risk of casualty crash doubles for every 5 kph over the speed limit.

FACT: In average conditions, a car travelling at 60 km/h will take about 45 metres to stop in an emergency braking situation. A car braking from 65 km/h will still be moving at close to 32 km/h after 45 metres travelled.

EXAMPLE 1. CRASH DATA INITIAL SPEED (kph)IMPACT SPEED (kph) DRY ROADWET ROAD 50Stops 55Stops14 60Touches

How much speed does the driver succeed in losing before impact? INITIAL SPEED (kph)IMPACT SPEED (kph) SPEED LOST (kph) 50Stops50 – 0 = – 14 = – 32 = – 44 = – 53 = – 63 = – 70 = 10

Each 5 kph added to the initial speed results in an approximately 20% decrease in the amount of speed the driver manages to lose by braking. WHY IS THIS???? INITIAL SPEED (kph) IMPACT SPEED (kph) SPEED LOST (kph) 50Stops50 – 0 = – 14 = – 32 = – 44 = – 53 = – 63 = – 70 = 10

Each 5 kph added to the initial speed results in an approximately 20% decrease in the amount of speed the driver manages to lose by braking. WHY IS THIS???? INITIAL SPEED (kph) IMPACT SPEED (kph) SPEED LOST (kph) 50Stops50 – 0 = – 14 = – 32 = – 44 = – 53 = – 63 = – 70 = % - 32% - 25% - 19% - 29% - 17%

EXAMPLE 2. PROFIT A company began doing business four years ago. Its profits for the last four years have been $11 million, $15 million, $ 19 million and $23 million. If the PATTERN continues the expected profit in 30 years is going to be $127 million WHY????

EXAMPLE 3. SQUARES & SQUARE NUMBERS HOW MANY POINTS WILL THE NEXT FIGURE HAVE? WHY??? WHAT ARE THESE SQUARES REPRESENTING????

All of the above are …

SEQUENCES

But … WHAT IS A SEQUENCE?

DEFINITION A SEQUENCE is a set of quantities arranged in a definite order. For example:  1, 2, 3, 4, 5, …  1, 4, 9, 16, 25, …  1, 8, 27, 64, 125, …  -10, -8, -6, -4, -2, …

TWO TYPES OF SEQUENCES 1. Arithmetic Sequence 2. Geometric Sequence

TWO TYPES OF SEQUENCES Arithmetic Sequence 1, 3, 5, 7, 9, … 11, 15, 19, 23, … Geometric Sequence 2, 6, 18, 54, , 20, 2, 0.2

How to distinguish an arithmetic series? An arithmetic sequence will always have a common difference between successive terms. For example:  2, 4, 6, 8, 10, …COMMON DIFFERENCE of 2  1, 4, 7, 11, 14, …COMMON DIFFERENCE of 3

GETTING BACK TO THE PROFIT EXAMPLE … How can you calculate the 27 th term? Moreover, how can you calculate the n th term? Tip: What is the common difference? YEARSPROFIT 111 mm 215 mm 319 mm 423 mm … 30????

GETTING BACK TO THE PROFIT EXAMPLE … The common difference is – 11 = 4 19 – 15 = 4 Therefore we know that we need to multiply the n th by 4 YEARSPROFIT 111 mm 215 mm 319 mm 423 mm … 30

GETTING BACK TO THE PROFIT EXAMPLE … But … 4(1) = 4, 4(2) = 8 and 4(3) = 12 … If we add + 7 we’ll get the result. Hence, in 30 years the profit will be 30(4) + 7 = YEARSPROFIT 111 mm 215 mm 319 mm 423 mm … mm

Tip: Look for it in your booklet!!!!!