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Number - Types of numbers and sequences

Grade 11IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Classification of Numbers: Natural numbers (N), Integers (Z), Rational numbers (Q), Irrational numbers, and Real numbers (R).

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Prime Numbers: Numbers with exactly two factors (1 and itself). 1 is not a prime number.

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Factors and Multiples: HCF (Highest Common Factor) and LCM (Lowest Common Multiple) using prime factorization.

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Arithmetic Sequences: A sequence where the difference between consecutive terms is constant (common difference, d).

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Geometric Sequences: A sequence where each term is found by multiplying the previous term by a constant (common ratio, r).

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Quadratic Sequences: A sequence where the second difference is constant. The n-th term is in the form an² + bn + c.

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Special Sequences: Square numbers (1, 4, 9...), Cube numbers (1, 8, 27...), and Fibonacci sequences (each term is the sum of the two preceding ones).

📐Formulae

Arithmetic n-th term: un=a+(n−1)du_n = a + (n - 1)d

Geometric n-th term: un=a×rn−1u_n = a \times r^{n-1}

Quadratic n-th term: un=an2+bn+cu_n = an^2 + bn + c

Second difference in quadratic sequence: 2a=constant second difference2a = \text{constant second difference}

Sum of terms (Arithmetic): Sn=n2(2a+(n−1)d)S_n = \frac{n}{2}(2a + (n-1)d)

💡Examples

Problem 1:

Find the n-th term of the sequence: 5, 8, 11, 14, ...

Solution:

un=3n+2u_n = 3n + 2

Explanation:

This is an arithmetic sequence with first term a=5a=5 and common difference d=3d=3. Using the formula un=a+(n−1)du_n = a + (n-1)d, we get un=5+(n−1)3=5+3n−3=3n+2u_n = 5 + (n-1)3 = 5 + 3n - 3 = 3n + 2.

Problem 2:

Determine the n-th term of the sequence: 4, 7, 12, 19, 28, ...

Solution:

un=n2+3u_n = n^2 + 3

Explanation:

First differences: 3, 5, 7, 9. Second differences: 2, 2, 2. Since the second difference is constant (2), it is a quadratic sequence. 2a=2⇒a=12a = 2 \Rightarrow a = 1. Comparing n2n^2 (1, 4, 9, 16) to our sequence (4, 7, 12, 19), each term is exactly 3 more than n2n^2.

Problem 3:

Is 25+π\sqrt{25} + \pi rational or irrational?

Solution:

Irrational

Explanation:

25=5\sqrt{25} = 5, which is a rational number. π\pi is a well-known irrational number (it cannot be written as a fraction p/qp/q). The sum of a rational and an irrational number is always irrational.

Problem 4:

Find the next term and the n-th term of the sequence: 2, 6, 18, 54, ...

Solution:

Next term: 162; n-th term: 2×3n−12 \times 3^{n-1}

Explanation:

This is a geometric sequence because each term is multiplied by 3 (common ratio r=3r = 3). The first term a=2a = 2. The next term is 54×3=16254 \times 3 = 162. The formula is un=arn−1u_n = ar^{n-1}.