Predicting What Comes Next: Sequences Class 9 Notes

Predicting What Comes Next: Exploring Sequences and Progressions is the Class 9 chapter on number patterns and how to continue them. This guide explains sequences, explicit and recursive rules, arithmetic progressions, the sum of the first n natural numbers, geometric progressions and fractals, with every worked example recomputed and important questions answered in full.

Sequences and terms

A sequence is an ordered list of numbers, and each number is a term. We write t₁ for the first term, t₂ for the second, and so on; the subscript is the position. The position n must be a positive integer, but the terms can be any real number, including fractions and negatives.

• Infinite sequence: goes on forever, shown by "...", like 1, 3, 5, ...
• Finite sequence: has a fixed number of terms, like 6, 12, 24, 48, 96.

For square numbers, t₅ = 5² = 25.

Explicit rules

An explicit rule gives a term straight from its position n, without earlier terms. It is the fast way to reach terms far down the list.

Finding a term: uₙ = 2n − 1. Then u₅₃ = 2(53) − 1 = 106 − 1 = 105.

Checking whether a number is a term: sₙ = 5n − 2. Is 471 a term? 5n − 2 = 471, so 5n = 473 and n = 94.6. Positions must be natural numbers (there is no 94.6th place), so 471 is not a term.

Recursive rules

A recursive rule builds each term from the one(s) before it. It needs two parts: a starting value and a rule for the next term.

Worked example: u₁ = 1, uₙ = 2uₙ₋₁ + 3 for n ≥ 2.

• u₂ = 2(1) + 3 = 5
• u₃ = 2(5) + 3 = 13
• u₄ = 2(13) + 3 = 29
• u₅ = 2(29) + 3 = 61

Some rules use two earlier terms. In the Virahānka-Fibonacci sequence, each term is the sum of the two before it: 1, 2, 3, 5, 8, 13, 21, 34, ... The next two terms are 21 + 34 = 55 and 34 + 55 = 89. Indian mathematicians like Virahānka studied it as early as the 7th century CE, in the context of poetry meters.

Arithmetic progressions (AP)

An AP adds the same number each time. That number is the common difference, d = tₙ − tₙ₋₁. It can be positive (increasing), negative (decreasing) or zero.

General form: a, a + d, a + 2d, a + 3d, ...

nth term: tₙ = a + (n − 1)d. You add d one time fewer than the position number.

Recursive form: if a = 5 and d = 3, then t₁ = 5 and tₙ = tₙ₋₁ + 3 for n ≥ 2.

If you plot position n against tₙ, the points of an AP always lie on a straight line.

Worked example 1: 3, 7, 11, ... has a = 3, d = 4. t₂₀ = 3 + 19 × 4 = 3 + 76 = 79.

Worked example 2: 20, 17, 14, ... has d = −3. t₁₀ = 20 + 9(−3) = 20 − 27 = −7.

Sum of the first n natural numbers

Write S = 1 + 2 + ... + n forwards and backwards, then add the two lines. You get n pairs, each adding to (n + 1). So 2S = n(n + 1), and:

Sₙ = n(n + 1)/2

This is also the rule for triangular numbers, and it appears in the Āryabhaṭīya by Āryabhaṭa.

• First 100 natural numbers: 100 × 101/2 = 50 × 101 = 5050.
• A range: 25 + 26 + ... + 58 = S₅₈ − S₂₄. S₅₈ = 58 × 59/2 = 1711 and S₂₄ = 24 × 25/2 = 300. Sum = 1711 − 300 = 1411.

Geometric progressions (GP)

A GP multiplies by the same non-zero number each time, the common ratio r.

General form: a, ar, ar², ar³, ...

nth term: tₙ = arⁿ⁻¹

GP terms grow or shrink very fast, and their graph is not a straight line. A fractional ratio makes terms shrink towards zero; a negative ratio makes them switch between positive and negative (for example 2, −6, 18, −54 with r = −3).

Worked example: Is 5, 15/4, 45/16, 135/64, ... a GP? (15/4) ÷ 5 = 3/4 and (45/16) ÷ (15/4) = (45/16) × (4/15) = 3/4. The ratio is constant, so r = 3/4 and tₙ = 5 × (3/4)ⁿ⁻¹.

For a = 2 and r = 3: t₄ = 2 × 3³ = 2 × 27 = 54.

Fractals and real-life uses

The Sierpiński triangle is made by repeatedly removing the middle of equilateral triangles.

Because r = 3/4 is less than 1, the shaded area gets closer and closer to 0.

Real-life models:

• AP: constant change, such as a taxi fare with a fixed booking fee plus a fixed rate per kilometre, or a salary with a fixed yearly increase. A salary starting at 5,00,000 and rising by 20,000 a year reaches 7,00,000 when 5,00,000 + (n − 1)20,000 = 7,00,000. So (n − 1)20,000 = 2,00,000, n − 1 = 10 and n = 11: in the 11th year.
• GP: a ball dropped from 24 ft that rises to 0.75 of its previous height bounces to 18 ft, then 13.5 ft, then 10.125 ft. Bacteria that start at 30 and double every hour number 30 × 2ⁿ after n hours.

Remember this

• Explicit rule: use n directly. Recursive rule: use the previous term.
• AP: tₙ = a + (n − 1)d. GP: tₙ = arⁿ⁻¹.
• If solving for n gives a non-natural number, the value is not a term.
• Sₙ = n(n + 1)/2; for a range, subtract the unwanted beginning.

Important questions with answers

1. Find the 6th term of the GP 3, 6, 12, ... a = 3, r = 2. t₆ = 3 × 2⁵ = 3 × 32 = 96.

2. Is 99 a term of 3, 7, 11, ...? 3 + (n − 1)4 = 99, so (n − 1)4 = 96, n − 1 = 24 and n = 25. Yes, it is the 25th term.

3. Is 100 a term of the same AP? (n − 1)4 = 97 gives n − 1 = 24.25, which is not a whole number. So 100 is not a term.

4. Find 11 + 12 + ... + 20. S₂₀ − S₁₀ = 210 − 55 = 155.

5. Find the sum of the first 50 natural numbers. 50 × 51/2 = 1275.

6. How many triangles are there at stage 4 of the Sierpiński triangle, and what is the shaded area? 3⁴ = 81 triangles; area = (3/4)⁴ = 81/256.

7. Find the 5th triangular number. 5 × 6/2 = 15. Check: 1 + 2 + 3 + 4 + 5 = 15.

Common mistakes to avoid

• Using n instead of (n − 1) in tₙ = a + (n − 1)d or in arⁿ⁻¹.
• Finding d as tₙ₋₁ − tₙ, which flips its sign.
• Accepting a fractional n as a valid position.
• Subtracting S₂₅ instead of S₂₄ when the range starts at 25.

For more practice on APs and GPs, study this chapter with Joy on Learnijoy.