Revision: Numbers and Sequences Mathematics SSLC (English Medium) Class 10 Tamil Nadu Board of Secondary Education

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

• 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.

• 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.

• Numbers in a sequence are called terms or elements.

• 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)

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 → S_n = na

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:

$0\le r<b$

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.

• Definition: An A.P. is a sequence in which the difference between consecutive terms is constant.
• Condition for A.P.: \[ a_{n + 1} - a_{n} = d \]
• General form: \[ a,\; a + d,\; a + 2d,\; a + 3d, \ldots \]