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Sequences and PatternsArithmetic and Geometric Sequences

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In 1682, the astronomer Edmond Halley observed an unusual phenomenon: a glowing white object with a long tail that moved across the night sky. It was a comet, a small, icy rock that is flying through space, while leaving behind a trail of dust and ice.

Halley remembered that other astronomers had observed similar comets much earlier: one in 1530 and another in 1606. Notice that the gap between two consecutive observations is the same in both cases: years.

Image of Halley’s Comet,
taken in 1986 on Easter Island

Halley concluded that all three observations were in fact of the same comet – which is now called Halley’s comet. It is orbiting around the sun and passes Earth approximately every 76 years. He also predicted when the comet would be visible next:

1530, 1606+76, 1682+76, 1758+76, +76, +76, +76, …

Actually, the time interval is not always exactly 76 years: it can vary by one or two years, as the comet’s orbit is interrupted by other planets. Today we know that Halley’s comet was observed by ancient astronomers as early as 240 BC!

Depections of Halley’s comet throughout time: a Babylonian tablet (164 BC), a medival tapestry (1070s), a science magazine (1910) and a Soviet stamp (1986).

A different group of scientists is investigating the behaviour of a bouncing tennis ball. They dropped the ball from a height of 10 meters and measured its position over time. With every bounce, the ball loses some of its original height:

The scientists noticed that the ball loses 20% of its height after every bounce. In other words, the maximum height of every bounce is 80% of the previous one. This allowed them to predict the height of every following bounce:

10, 8×0.8, ×0.8, ×0.8, 4.096×0.8, 3.277×0.8, 2.621×0.8, 2.097×0.8, …

Definitions

If you compare both these problems, you might notice that there are many similarities: the sequence of Halley’s comet has the same differenceratioproduct between consecutive terms, while the sequence of tennis ball bounces has the same ratiodifferenceproduct between consecutive terms.

Sequences with these properties have a special name:

An arithmetic sequence has a constant difference d between consecutive terms.

The same number is added or subtracted to every term, to produce the next one.

A geometric sequence has a constant ratio r between consecutive terms.

Every term is multiplied or divided by the same number, to produce the next.

Here are a few different sequences. Can you determine which ones are arithmetic, geometric or neither, and what the values of d and r are?

2, 4, 8, 16, 32, 64, …

is geometricarithmeticneither, with ratio .

2, 5, 8, 11, 14, 17, …

is arithmeticgeometricneither, with difference .

17, 13, 9, 5, 1, –3, …

is arithmeticgeometricneither, with difference .

2, 4, 7, 11, 16, 22, …

is neitherarithmeticgeometric.

40, 20, 10, 5, 2.5, 1.25, …

is geometricarithmeticneither, with ratio .

To define an arithmetic or geometric sequence, we have to know not just the common difference or ratio, but also the initial value (called a). Here you can generate your own sequences and plot their values on a graph, by changing the values of a, d and r. Can you find any patterns?

Arithmetic Sequence

a = ${a}, d = ${d}


${arithmetic(a,d,0)}, ${arithmetic(a,d,1)}, ${arithmetic(a,d,2)}, ${arithmetic(a,d,3)}, ${arithmetic(a,d,4)}, ${arithmetic(a,d,5)}, …

Geometric Sequence

a = ${b}, r = ${r}


${geometric(b,r,0)}, ${geometric(b,r,1)}, ${geometric(b,r,2)}, ${geometric(b,r,3)}, ${geometric(b,r,4)}, ${geometric(b,r,5)}, …

Notice how all arithmetic sequences look very similar: if the difference is positive, they steadily increasedecrease, and if the difference is negative, they steadily decreaseincrease.

Geometric sequences, on the other hand, can behave completely differently based on the values of a and r:

If r>1, the terms will quickly get biggerquickly decreaseget closer to zero, up to infinity. Mathematicians say that the sequence diverges.

If r is between –1 and 1, the terms will always get closer to 0decrease to negative infinityget smaller. We say that the sequence converges.

If r<1, the terms will alternate between positive and negative, while their absolute valueinversedifference gets bigger.

You’ll learn more about convergence and divergence in the last section of this course.

Recursive and Explicit Formulas

In the previous section, you learned that a recursive formula tells you the value of each term as a function of previous terms. Here are the recursive formulas for arithmetic and geometric sequences:

xn= xn1+dxn1×dxd+n

xn= xn1×rxn1rxnr+n

One problem with recursive formulas is that to find the 100th term, for example, we first have to calculate the previous 99 terms – and that might take a long time. Instead, we can try to find an explicit formula, that tells us the value of the nth term directly.

For arithmetic sequences, we have to add d at every step:

x1= a

x2= a+d

x3= a+d+d

x4=

x5=

At the nth term, we are adding n1nn+1 copies of d, so the general formula is

xn=a+d×n1.

For geometric sequences, we have to multiply r at every step:

x1=a

x2=a×r

x3=a×r×r

x4=

x5=

At the nth term, we are multiplying n1nn+1 copies of r, so the general formula is

xn=a×rn1.

Here is a summary of all the definitions and formulas you’ve seen so far:

An arithmetic sequence has first term a and common difference d between consecutive terms.

Recursive formula: xn=xn1+d

Explicit formula: xn=a+d×n1

A geometric sequence has first term a and common ratio r between consecutive terms.

Recursive formula: xn=xn1×r

Explicit formula: xn=a×rn1

Now let’s have a look at some examples where we can use all this!

Pay it Forward

Here is a short clip from the movie Pay it Forward, where 12-year-old Trevor explains his idea for making the world a better place:

Extract from “Pay It Forward” (2000), © Warner Bros. Entertainment

The essence of Trevor’s idea is that, if everyone “pays it forward”, a single person can have a huge impact on the world:

Notice how the number of people at every step forms a geometric sequencearithmetic sequencetriangle number, with common ratio :

1, 3×3, 9×3, ×3, ×3, ×3, …

Using the explicit formula for geometric sequences, we can work out how many new people are affected at any step:

xn =

The number of people increases incredibly quickly. In the 10th step, you would reach 19,683 new ones, and after 22 steps you would have reached more people than currently alive on Earth.

This sequence of numbers has a special name: the powers of 3. As you can see, every term is actually just a different power of 3:

30, 31, 32, 33, 34, 35, …

Who wants to be a Millionaire?

COMING SOON!

The Chessboard Problem

COMING SOON!