Definitions [6]
A set of numbers where the numbers are arranged in a definite order, like the natural numbers, is called a sequence.
An Arithmetic Progression (A.P.) is a sequence in which the difference between consecutive terms is constant.
- Common difference = d = second term − first term
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The general form of an AP is a, a + d, a + 2d, a + 3d, …
a = first term
d = common difference
When the numbers (terms) in a sequence are connected to each other by a positive (plus) sign or a negative (minus) sign, the sequence becomes a series.
A progression is a sequence where each term follows a uniform rule.
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Every progression is a sequence, but with a clear pattern.
A sequence is a group of numbers arranged in a definite order following a rule.
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Numbers in a sequence are called terms or elements.
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The term at position n is called the nth term, denoted by Tₙ.
A sequence, in which each of its terms can be obtained by multiplying or dividing its preceding term by a fixed quantity, is called a geometric progression.
- A fixed number is called the common ratio (r)
Formulae [1]
If r < 1→ use \[S_n=\frac{a(1-r^n)}{1-r}\]
If r > 1 → use \[S_n=\frac{a(r^n-1)}{r-1}\]
If r = 1 → Sn = na
Theorems and Laws [2]
Statement:
Euclid’s Division Algorithm states that for any two positive integers a and b, there exist whole numbers q and r such that when a is divided by b, the remainder r is smaller than b.
Equation:
a = bq + r,
Statement:
Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
