Sequences and PatternsArithmetic and Geometric Sequences

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!

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:

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?

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?

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:

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:

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.

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

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:

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 are 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?

Aarav and Diya are negotiating their pocket money with their parents, in exchange for doing chores around the house. They compare two different payment systems:

₹10 sounds much better than ₹1, so Aarav is sure that he got the better deal. Diya just smiles. Who do you think will have earned more after one month?

Let’s compare the two payments side by side, day by day. Can you fill in the missing amounts?

Aarav’s payments form an arithmetic sequence: the first term is 10, and the difference is . Diya’s payments form a geometric sequence: the first term is 1, and the ratio is .

Aarav’s income grows linearly, while Diya’s income doubles every day. Which one would you pick?

On day 5, Aarav is still well ahead: ₹50, compared to just ₹16 for Diya. But the gap is closing fast. Diya’s payment on day n is

xn =

Let’s continue both sequences for a few more days:

Aarav: 90, 100 +10, 110, 120, …

Diya: 256, 512 ×2, 1024, , 4096, …

On day 12, Diya earns ₹2,048 in a single day – about 17 times Aarav’s ₹120. And from now on, the gap doubles every day!

Here you can compare any arithmetic and geometric sequence, by changing their first term a, Aarav’s difference d, and Diya’s ratio r. No matter how large a and d are: can you find any values that let Aarav win in the long run?

As long as the ratio is bigger than 1, a geometric sequence will eventually overtake every arithmetic sequence – even one that starts with a million rupees and grows by a million every day. This explosive pattern is called exponential growth.

Diya’s daily payment is 2n−1, and it grows at a breathtaking speed:

After just days, Diya earns ₹512 in a single day. On day 15, she receives 214 = ₹16,384 – while Aarav is still stuck at ₹150.

On day , Diya’s payment reaches 220 = ₹1,048,576. She has become a millionaire – in less than one month, starting from a single rupee! Aarav, meanwhile, has earned just ₹2,310 in total.

Diya’s total earnings after 21 days are 221−1 = ₹2,097,151 – more than double a million. (You’ll learn a clever trick for calculating such sums in the next section.)

The Chessboard Problem

The game of chess was invented in India, many hundreds of years ago. According to legend, the king loved the new game so much that he summoned its inventor to his palace and promised him any reward he wanted.

The inventor’s request sounded surprisingly modest: rice. Place one grain of rice on the first square of the chessboard, two grains on the second square, four grains on the third, and so on – every square gets twice as many grains as the one before.

Move the slider to fill the chessboard, square by square, and watch what happens to the number of grains:

The number of grains on each square forms a geometric sequence: the first term is , and the common ratio is .

That means the number of grains on square n is xn=2n−1 – exactly like Diya’s pocket money in the previous section.

The 10th square already needs grains, and the 20th square needs 524,288 – more than half a million. And the board has 64 squares!

The king doesn’t just pay for the last square – he has to pay for every square. So the total number of grains is the sum of the whole sequence:

S=1+2+4+8+…+262+263

Here is a clever trick: multiply the entire sum by the common ratio, 2164:

2S=2+4+8+…+263+264

Almost every term of 2S also appears in S, just one position later. If we subtract the first equation from the second one, almost everything cancels:

2S−S=264−1

And therefore:

S= 264−1264263

That is 18,446,744,073,709,551,615 grains of rice – roughly 18 billion billion!

The same trick works for any geometric sequence: the sum of the first n terms with ratio r is

Sn=rn−1r−1

So how much rice is that, really? A single grain weighs about 0.025 grams, which means the king’s debt is around 460 billion tonnes of rice – almost a thousand years of the entire world’s rice harvests.