01 · Explore
Introduction to Sequences
A sequence is an ordered list of numbers where each number is called a term. Patterns in sequences help us make sense of the world and predict future values.
In mathematics, we encounter various types of sequences. For example, natural numbers (1, 2, 3, ...), odd numbers (1, 3, 5, ...), and square numbers (1, 4, 9, ...) are all infinite sequences. The three dots (...) at the end indicate that the sequence continues indefinitely. Conversely, a sequence like 6, 12, 24, 48, 96 is finite because it has a specific number of terms.
We use notation to identify the position of a term. For a sequence 't', t₁ represents the first term, t₂ the second, and so on. The subscript always matches the position number. While the position 'n' must be a positive integer (1, 2, 3...), the terms themselves can be any real number, including fractions or negative integers.
| Sequence Type | First Four Terms | Pattern Description |
|---|---|---|
| Natural Numbers | 1, 2, 3, 4 | Each term is 1 more than the previous. |
| Odd Numbers | 1, 3, 5, 7 | Each term is 2 more than the previous. |
| Square Numbers | 1, 4, 9, 16 | Terms are squares of their position (n²). |
| Triangular Numbers | 1, 3, 6, 10 | Sum of natural numbers up to that position. |
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Question
In the sequence of square numbers, what is the value of t₅?
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02 · Explore
Explicit Rule for a Sequence
An explicit rule is a formula that uses a term's position number, n, to calculate its value directly without needing previous terms.
Explicit formulas are highly efficient for finding terms deep in a sequence. For example, if we want to find the 1000th term of a sequence, we simply substitute n = 1000 into the formula. This saves us from having to calculate the first 999 terms.
We can also use explicit rules to verify if a specific number belongs to a sequence. By setting the formula equal to the number and solving for 'n', we can determine its position. If 'n' results in a natural number, the value is a term; if 'n' is a fraction or decimal, the value is not part of that sequence.
Finding a Specific Term
uₙ = 2n - 1; Find u₅₃
Substitute n = 53 into the formula: u₅₃ = 2(53) - 1 = 106 - 1 = 105. The 53rd term is 105.
Checking if a Number is a Term
sₙ = 5n - 2; Is 471 a term?
Set 5n - 2 = 471. Adding 2 to both sides gives 5n = 473. Dividing by 5 gives n = 94.6. Since 94.6 is not a natural number, 471 is not a term in this sequence.
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Why must 'n' be a natural number when solving for a term's position?
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03 · Explore
Recursive Rule for a Sequence
A recursive rule defines a term based on its relationship to the term(s) immediately preceding it.
Unlike explicit rules, a recursive rule requires you to know the previous term to find the next one. It usually consists of two parts: the starting value (like t₁) and the rule for subsequent terms (like tₙ = tₙ₋₁ + 3).
Some recursive rules involve more than one previous term. The most famous example is the Virahānka-Fibonacci sequence, where each term is the sum of the two preceding terms. This sequence (1, 2, 3, 5, 8, 13...) was studied by Indian mathematicians like Virahānka as early as the 7th century CE in the context of poetry meters.
Generating Terms Recursively
u₁ = 1, uₙ = 2uₙ₋₁ + 3 for n ≥ 2
To find u₂: 2(u₁) + 3 = 2(1) + 3 = 5. To find u₃: 2(u₂) + 3 = 2(5) + 3 = 13. To find u₄: 2(u₃) + 3 = 2(13) + 3 = 29.
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Question
What are the next two terms of the Virahānka-Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34...?
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04 · Explore
Arithmetic Progressions (AP)
An Arithmetic Progression is a sequence where the difference between any two consecutive terms is a constant value, called the common difference (d).
In an AP, we start with a first term 'a' and add the common difference 'd' repeatedly. The general form is a, a+d, a+2d, a+3d... This leads to the explicit formula tₙ = a + (n - 1)d. Note that 'd' can be positive (increasing sequence), negative (decreasing sequence), or even zero.
The common difference 'd' is specifically defined as the difference between a term and its predecessor: d = tₙ - tₙ₋₁. When we plot the terms of an AP on a graph where the x-axis is the position (n) and the y-axis is the term value (tₙ), the points always lie on a straight line. This linear relationship is a defining visual characteristic of arithmetic progressions.
Structure of an AP Term
- 1
First Term (a)
The starting value of the sequence at position n=1.
- 2
Common Difference (d)
The fixed amount added to each term to get the next (tₙ - tₙ₋₁).
- 3
Position (n-1)
We add the difference one less time than the position number.
The formula tₙ = a + (n-1)d combines these elements to find any term directly.
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If an AP has a = 5 and d = 3, what is the recursive rule?
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Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 7, 8, 9, 10.