Mathematics Notes: Number Systems & Series
1. Number Systems
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Prime Numbers:
- Definition: A natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself.
- Examples: 2, 3, 5, 7, 11, 13, ...
- Hint: 2 is the only even prime number.
- Hint: 1 is neither prime nor composite.
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Composite Numbers:
- Definition: A natural number greater than 1 that has more than two distinct positive divisors.
- Examples: 4, 6, 8, 9, 10, 12, ...
- Hint: 1 is neither prime nor composite.
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Co-prime Numbers (or Relatively Prime Numbers):
- Definition: Two numbers are co-prime if their Highest Common Factor (HCF) or Greatest Common Divisor (GCD) is 1. They share no common prime factors.
- Examples: (4, 9), (7, 15), (8, 25)
- Hint: Co-prime numbers do not have to be prime themselves.
2. Consecutive Numbers and Averages
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Representing Consecutive Numbers:
- If 'A' is the first number, then consecutive integers are: A, A+1, A+2, A+3, ...
- Consecutive even numbers are: A, A+2, A+4, A+6, ... (where A is even)
- Consecutive odd numbers are: A, A+2, A+4, A+6, ... (where A is odd)
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Finding the Average and Terms:
- Hint (Odd count of terms): If you have an odd number of consecutive integers (or consecutive even/odd integers), their average is always the middle number in the sequence.
- Hint (Even count of terms): If you have an even number of consecutive integers, their average is always the value exactly halfway between the two middle numbers.
- General Method: (Sum of terms) / (Number of terms) = Average. You can set up an equation by representing the terms using a single variable (e.g., 'a') and solve for 'a'.
3. Summation Formulas (Specific Series)
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Sum of the first n Natural Numbers:
- Formula: S_n = n(n+1)/2
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Sum of the first n Even Numbers:
- Formula: S_n = n(n+1)
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Sum of the first n Odd Numbers:
- Formula: S_n = n^2
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Sum of the Squares of the first n Natural Numbers:
- Formula: S_n = n(n+1)(2n+1)/6
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Sum of the Cubes of the first n Natural Numbers:
- Formula: S_n = (n(n+1)/2)^2
- OR S_n = (Sum of the first n natural numbers)^2
4. Arithmetic Progressions (AP)
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Definition: A sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference (d).
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Common Difference (d):
- Formula: d = a2 - a1 (any term minus its preceding term)
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nth term (an) of an AP:
- Formula: an = a1 + (n-1)d
- Where:
- a_n = the nth term
- a_1 = the first term
- n = the number of terms (or position of the term)
- d = the common difference
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Sum of the first n terms (Sn) of an AP:
- Formula 1 (if a1 and an are known): Sn = n/2 * (a1 + an)
- Formula 2 (if a1 and d are known): Sn = n/2 * (2*a1 + (n-1)d)
5. Quick Identification of Co-prime Numbers (Exam Strategy)
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Definition: HCF (or GCD) is 1.
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Quick Elimination Checks (If they share any of these, they are NOT co-prime):
- Hint: If both numbers are even, they are not co-prime (HCF is at least 2).
- Hint: If the sum of digits for both numbers is divisible by 3, they are not co-prime (HCF is at least 3).
- Hint: If both numbers end in 0 or 5, they are not co-prime (HCF is at least 5).
- Hint: If both numbers end in 0, they are not co-prime (HCF is at least 10).
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Mental Euclidean Algorithm (for less obvious cases):
- The principle is HCF(a, b) = HCF(a, b-a) for b > a.
- Hint: Repeatedly replace the larger number with the difference between the two numbers (e.g., HCF(72, 108) -> HCF(72, 36)). Keep simplifying until you reach a point where the common factor is obvious or you confirm the HCF is 1 (often when one number becomes prime and the other is not its multiple).
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Key Hints (Always Co-prime):
- Hint: Any two consecutive integers are always co-prime (e.g., 10, 11).
- Hint: Any two distinct prime numbers are always co-prime (e.g., 17, 23).
- Hint: A prime number and any other number that is NOT its multiple are always co-prime (e.g., 11 and 30, since 30 is not a multiple of 11).
