Predicting What Comes Next: Exploring Sequences and Progressions

Why This Matters

Look around you. Patterns are everywhere.

The seats in a cinema hall grow by a fixed number in each new row. Your savings grow by a fixed amount every month if you put aside the same pocket money. A taxi fare goes up by the same rupees for every extra kilometre. Even the months of the year repeat in a pattern.

Our brains love patterns. The moment we spot one, we can predict what comes next. If a row of tiles goes 2, 5, 8, 11, …, you already feel the next number should be 14. You did not check anything. You just sensed the rhythm.

This chapter turns that feeling into a tool. We will look at lists of numbers that follow a rule, called sequences. We will find neat formulas that let us jump straight to the 100th number — or the 1000th — without writing out all the ones before it. And we will meet a special, very common kind of sequence called an arithmetic progression, the workhorse behind salaries, savings, fares and seating.

By the end, you will not just guess what comes next. You will be able to prove it, for any position you like.

The Big Idea

The Big Idea: A sequence is just an ordered list of numbers that follows a rule. Once you find the rule, you can predict any term you want. The most powerful rules are explicit formulas — they take a position number n and hand you that term directly, with no need to know the terms before it. For a list that grows by a fixed step, that formula is short and beautiful, and a clever trick even lets us add up a whole list in one line.

Let’s Break It Down

We will build this up in steps. First, what a sequence is and how we name its terms. Then two ways to describe a sequence — an explicit rule and a recursive rule. Then the star of the chapter, the arithmetic progression (AP), with its nth-term formula. Then a famous summing trick that adds 1 + 2 + … + n in one move. Finally, a cousin of the AP called the geometric progression (GP).

Let us start with some sequences you have already met in earlier classes.

What Is a Sequence?

You have seen these lists before:

• Natural numbers: 1, 2, 3, 4, 5, 6, …
• Odd numbers: 1, 3, 5, 7, 9, 11, …
• Triangular numbers: 1, 3, 6, 10, 15, 21, …
• Square numbers: 1, 4, 9, 16, 25, 36, …

The three dots … mean the list keeps going forever.

A sequence is an ordered list of numbers. Each number in the list is called a term.

The word ordered matters. In a sequence, position is everything. In the square numbers, 1 is the first term, 4 is the second term, and 25 is the fifth term. Swapping them would make a different sequence.

A sequence can be finite (it stops) or infinite (it never stops). The four lists above are infinite. But the list 6, 12, 24, 48, 96 is finite — it has exactly five terms and then it ends. The days of one week, or the overs in one cricket innings, are finite sequences from daily life.

Triangular numbers: a picture of the pattern

Let us look closely at the triangular numbers: 1, 3, 6, 10, 15, …

Why “triangular”? Because each one can be drawn as a triangle of dots. And there is a lovely pattern hiding inside. Watch:

• 1 = 1
• 3 = 1 + 2
• 6 = 1 + 2 + 3
• 10 = 1 + 2 + 3 + 4
• 15 = 1 + 2 + 3 + 4 + 5

So the nth triangular number is just the sum of the first n natural numbers. The picture below makes this obvious.

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Notice the differences between consecutive terms: from 1 to 3 is +2, from 3 to 6 is +3, from 6 to 10 is +4, from 10 to 15 is +5. The jumps themselves grow. So the triangular numbers do not grow by a fixed step. Keep that in mind — it will matter soon.

Square numbers: hidden odd numbers

Now the square numbers: 1, 4, 9, 16, 25, 36, …

Their consecutive differences are 3, 5, 7, 9, 11 — all the odd numbers! And just like before, each square is a running sum:

• 1 = 1
• 4 = 1 + 3
• 9 = 1 + 3 + 5
• 16 = 1 + 3 + 5 + 7
• 25 = 1 + 3 + 5 + 7 + 9

So every square number is the sum of the odd numbers up to that point. This is not a coincidence. The picture below shows exactly why.

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Naming the terms

Writing “the fourth term” again and again is clumsy. So we use a short notation. We write t₁ for the first term, t₂ for the second term, t₃ for the third, and in general tₙ for the term in position n.

The little number below — the subscript — is the position. So for the odd numbers, t₁ = 1, t₂ = 3, t₃ = 5, t₄ = 7. Reading t₄ = 7 out loud: “the term in the 4th position is 7.”

If we ever talk about two sequences at once, we use different letters: t₁, t₂, t₃, … for one, s₁, s₂, s₃, … for another, u₁, u₂, u₃, … for a third.

One careful point. The position n is always a counting number (1, 2, 3, …). But a term itself can be anything — a fraction, a negative number, any real number. For example the sequence –7, –3, 1, 5, 9, … has s₁ = –7, and each term is 4 more than the one before. The terms go negative, but the positions stay 1, 2, 3, ….

Two Ways to Describe a Sequence

There are two main ways to write down the rule for a sequence. They feel very different, so it is worth seeing both clearly.

The explicit rule (jump straight to any term)

An explicit rule gives the nth term directly from its position n. You plug in n and out comes the term — no need to know any other term.

Take the rule uₙ = 2n – 1. Substitute positions one by one:

• u₁ = 2 × 1 – 1 = 1
• u₂ = 2 × 2 – 1 = 3
• u₃ = 2 × 3 – 1 = 5

These are the odd numbers. So uₙ = 2n – 1 is the explicit rule for the odd-number sequence.

Why is this so useful? Because you can leap straight to any term. Want the 300th odd number? Just compute u₃₀₀ = 2 × 300 – 1 = 599. You do not need the 299 terms before it.

The explicit rule also lets us check if a number belongs to a sequence, and find its position. Is 137 an odd number, and if so, which one? Solve 2n – 1 = 137. That gives 2n = 138, so n = 69. Since 69 is a whole number, 137 is indeed the 69th term.

This relies on solving a simple equation, which you met in earlier classes. Here is a quick refresher.

Let us put that to work on a fuller example.

The recursive rule (build each term from the one before)

The second way is a recursive rule. It tells you the first term, then a rule to get each new term from the term(s) before it.

Take the sequence 1, 4, 7, 10, 13, …. Each term is 3 more than the previous one. So:

• t₂ = t₁ + 3
• t₃ = t₂ + 3
• t₄ = t₃ + 3

In general, tₙ = tₙ₋₁ + 3. Together with the starting value t₁ = 1, this completely describes the sequence. (Here tₙ₋₁ just means “the term one place before tₙ”.)

The recursive rule has a cost. To find the 100th term, you must climb up through all 99 earlier terms one by one. With an explicit rule you could skip straight there. But the recursive rule is often the most natural way to describe a pattern, because it captures exactly how the pattern grows. Both descriptions are useful; they are two views of the same sequence.

Let us see a recursive rule that is harder to turn into an explicit one.

A recursive rule does not have to use only the previous term. It can use the previous two terms. The most famous example starts with two given terms and adds them to make the next:

V₁ = 1, V₂ = 2, and Vₙ = Vₙ₋₁ + Vₙ₋₂ for n ≥ 3.

Working it out: V₃ = 2 + 1 = 3, V₄ = 3 + 2 = 5, V₅ = 5 + 3 = 8. This gives 1, 2, 3, 5, 8, 13, 21, 34, … — the Virahānka–Fibonacci sequence. It was written down and studied by the Indian scholar Virahānka in the 7th century CE, while studying the rhythms of Prakrit poetry, long before the Italian Fibonacci met it around 1200 CE. It shows up all over nature and mathematics.

Arithmetic Progressions (AP)

Now we meet the most useful special kind of sequence.

Look at the growing pattern of squares in Figure 8.3. Counting the tiny squares at each stage gives 1, 5, 9, 13, ….

Tap to zoom

Look at how the count is built: 1, then 1 + 4, then 1 + 4 + 4, then 1 + 4 + 4 + 4. We can rewrite this neatly:

1, 1 + 1 × 4, 1 + 2 × 4, 1 + 3 × 4, …

So the number of squares always jumps by the same fixed amount, 4. A sequence where the difference between consecutive terms is constant has a special name.

An arithmetic progression (AP) is a sequence in which each term after the first is found by adding a fixed number d to the previous term. The fixed number d is called the common difference.

Let us name the parts. The first term is called a. The common difference is d (the constant step). For the pattern above, a = 1 and d = 4.

The common difference can be negative too. In 11, 7, 3, –1, –5, …, each term is 4 less than the one before. This is still an AP — here a = 11 and d = –4. A negative d simply means the sequence goes down by a fixed step.

So the general AP looks like this:

a, a + d, a + 2d, a + 3d, …

Deriving the nth-term formula

Now the big payoff. We want a formula for the nth term of any AP, so we can jump straight to the 50th or 500th term.

Look again at the general AP. The first term is a. To reach the second term we add d once. To reach the third term we add d twice. To reach the fourth term, three times. Do you see the pattern? To reach the nth term, we add d exactly (n – 1) times — one time fewer than n. The picture below shows why.

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So we can write the nth-term formula:

tₙ = a + (n – 1) × d

where a is the first term, d is the common difference, and n is the position. This is the explicit rule for any AP. Memorise this one — you will use it constantly.

Let us check it on our pattern. For 1, 5, 9, 13 we had a = 1, d = 4. So tₙ = 1 + (n – 1) × 4 = 1 + 4n – 4 = 4n – 3. Test it: t₃ = 4 × 3 – 3 = 9. Correct.

The recursive rule for the same AP is short and natural too:

t₁ = a

tₙ = tₙ₋₁ + d for n ≥ 2

In words: start at a, then keep adding d.

Let us use the formula on a real problem.

The formula also works backwards — to find which position a number sits in.

Why an AP makes a straight line

Here is a beautiful fact. If we plot an AP — putting the position n on the x-axis and the term value on the y-axis — the points land exactly on a straight line.

Why? Because each time n goes up by 1, the value goes up by the same fixed d. Equal steps across, equal steps up. That is precisely what a straight line is. Figure 8.5 shows this for our 1, 5, 9, 13, 17 sequence.

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An AP from everyday life

APs are everywhere once you look. Here is a taxi fare.

The Sum of the First n Natural Numbers

Now a different but related question. How do we add up a long list quickly?

Try this: what is 1 + 2 + 3 + … + 100? Adding one at a time would take ages. There is a famous trick, said to have been used by the young mathematician Gauss. It is wonderfully simple.

Gauss’s pairing trick

Let S stand for the sum we want. Write it out, and then write it again backwards underneath:

S = 1 + 2 + 3 + 4 + 5 + 6

S = 6 + 5 + 4 + 3 + 2 + 1

Now add the two lines column by column. Look at each column: 1 + 6 = 7, 2 + 5 = 7, 3 + 4 = 7, and so on. Every column adds to the same number, 7! Figure 8.6 shows this clearly.

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We have six columns, each adding to 7. So adding the two copies of S together gives:

2S = 6 × 7 = 42

S = 42 ÷ 2 = 21

And indeed 1 + 2 + 3 + 4 + 5 + 6 = 21. The trick worked.

The general formula

The same idea works for any n. Write the sum forwards and backwards:

S = 1 + 2 + … + n

S = n + (n − 1) + … + 1

Each of the n columns adds to (n + 1). So 2S = n(n + 1), which gives:

Sₙ = n(n + 1) / 2

This is the formula for the sum of the first n natural numbers. The earliest known written statement of this result is in Āryabhaṭa’s Āryabhaṭīya (Chapter 2, Verse 19), from the 5th century CE.

Let us answer the original question with it. For 1 + 2 + … + 100, take n = 100: S₁₀₀ = 100 × 101 / 2 = 10100 / 2 = 5050. One line, done.

Notice something neat: this is exactly the formula for the triangular numbers from Figure 8.1, because the nth triangular number is the sum 1 + 2 + … + n. So tₙ = n(n + 1) / 2 gives you triangular numbers too.

The formula even lets us add up a partial run, like 25 + 26 + … + 58. The trick: take the whole sum up to 58 and subtract the part you do not want (up to 24).

Geometric Progressions (GP)

There is one more special sequence worth meeting. In an AP we add the same number each time. In a geometric progression, we multiply by the same number each time.

Look at another growing pattern of squares (Figure 8.6 in your textbook): the green-square counts go 3, 6, 12, 24. Each stage doubles the one before. So:

3, 3 × 2, 3 × 4, 3 × 8, … which is 3, 3 × 2, 3 × 2², 3 × 2³, …

A geometric progression (GP) is a sequence in which each term after the first is found by multiplying the previous term by a fixed number. That fixed number is called the common ratio, written r.

For 3, 6, 12, 24, the common ratio is r = 2, because 6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2 — the ratio of any term to the one before is always 2.

The general GP looks like this:

a, ar, ar², ar³, …

and the nth term is:

tₙ = a × rⁿ⁻¹

(The power is n − 1 for the same reason as in an AP: to reach the nth term you multiply by r one time fewer than n.)

A GP grows much faster than an AP. And unlike an AP, its plotted points do not lie on a straight line — they curve upward steeply. Figure 8.7 puts the two side by side.

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Let us check whether a given sequence is a GP.

GPs appear in many natural patterns — for example fractals like the Sierpiński triangle, where the number of small triangles triples at each stage (1, 3, 9, 27, …, a GP with r = 3). We will explore these more in later grades.

Common Mistakes

Even when the idea is clear, a few traps catch students often. Here are the big ones.

Quick Check

Test yourself before moving to practice.

Practice Problems

Try each one yourself first, then reveal the full solution.

Easy

Medium

Challenge

Summary

After working through this chapter, you can now explain:

• What a sequence is — an ordered list of numbers, each one a term — and how to name terms with the notation t₁, t₂, …, tₙ.
• The difference between an explicit rule (gives any term straight from its position n) and a recursive rule (builds each term from earlier terms), and when each is handy.
• What an arithmetic progression (AP) is — a sequence with a constant common difference d — and how to spot one by checking that consecutive differences are equal.
• Why the nth term of an AP is tₙ = a + (n − 1) × d, including why it is (n − 1) and not n: you count the jumps between terms, not the terms themselves.
• How to use Gauss’s pairing trick to show Sₙ = n(n + 1) / 2 for the sum of the first n natural numbers, and to add partial runs of numbers.
• What a geometric progression (GP) is — a sequence with a constant common ratio r, where you multiply instead of add — and its nth term tₙ = a × rⁿ⁻¹, and why a GP curves up while an AP makes a straight line.

What’s Next

That completes Part I of Ganita Manjari. Part II continues the journey, with more geometry, more algebra, and ways to handle and read data. The sequences and progressions you just learned will return again and again — in growth problems, in patterns, and later in deeper ideas about series. For now, head back to Class 9 Maths — Part I to revisit any chapter and lock in what you have learned.