Sequence and Series
* A collection of numbers arranged in a defined order according to some definite rule is called a sequence.
* The members or numbers that are listed in a sequence are called its terms.
* If the number of terms in a sequence is finite or countable, then it is called a finite sequence.
* If the sequence goes on forever or has an uncountable number of terms, then it is called an infinite sequence.
* The expression obtained on adding the terms of a sequence is called a series.
* A series is finite if the sequence is finite and infinite if the sequence is infinite.
Arithmetic Progression
* A collection of numbers arranged in a definite order according to some definite rule or pattern is called a sequence.
* A sequence is said to be in arithmetic progression, if each term (except the first) is obtained by adding a fixed number to its preceding term.
* The nth or general term of an AP is t_{n} = a + (n – 1)d.
* For an AP: a, a + d, a + 2d, a + 3d, a + 4d …. S_{n} = \frac{n}{2} [2a + (n – 1)d] or S_{n} = \frac{n}{2} [a + l], where l = a + (n – 1)d.
* If a constant is added to each term of an AP, then the resulting sequence is also an AP.
* If the constant is subtracted from each term of an AP, then the resulting sequence is also an AP.
* If each term of an AP is multiplied by a constant, then resulting sequence is also an AP.
* If each term on an AP is divided by a non-zero constant, then the resulting sequence is also an AP.
* If a constant is subtracted from each term of an AP, then the resulting sequence is also an AP.
* If a1, a2, a3, … and b1, b2, b3, … are two AP’s, then a1 + b1, a2 + b2, a3 + b3, … are also in AP.
* If a1, a2, a3, … and b1, b2, b3, … are two AP’s then a1 – b1, a2 – b2, a3 – b3,… are also in AP, if and only if the common difference of the two given sequences is not
the same.
Sum of First n Terms of an AP
Sum of n terms in an A.P is S = \frac{n}{2} [2a + (n – 1)d].
Geometric Progressions
* A sequence is said to be a geometric progression (GP), if each term (except the first) is obtained by multiplying its preceding term by a noon-zero constant.
* The common ratio, ‘r,’ is defined as the ratio of any term (except the first) to its preceding term, and is expressed as r = \frac{ a_{n + 1} }{ a_{n} } ; ∀ n > 1.
The nth term of a GP is an = arn-1.
The sum of n terms of GP is
S_{n} = \frac{ a ( r^{n}) }{ ( r - 1) } (r > 1) where (r > 1) = \frac{ a ( r^{n}) }{ ( r - 1) } (r < 1)
A finite GP → a, ar, ar2, ar3, …, arn-1
An infinite GP → a, ar, ar2, ar3, …
a + ar + ar2+ ar3 + … + arn-1 is called a finite GP series.
a + ar + ar2+ ar3+ …. is called an infinite GP series.
Geometric Mean
* The geometric mean (GM) between two number is number, which, when placed between them, forms with them a geometric progression.
* The geometric mean between a and b is \sqrt{ab}.
* The arithmetic mean (AM) between two numbers is a number, which, when placed between them, forms with them an arithmetic progression.
* The AM between two positive real numbers is always greater than or equal to their GM.
Sum of n Terms of Special Series
* The sum of the first n natural numbers,
1 + 2 + 3 +… + is \frac{n(n + 1)}{2}.
* The sum of the squares of the first n natural numbers,
12+ 22+32+… + n2 is \frac{n(n + 1) (2n + 1)}{6}.
* The sum of the cubes of the first n natural numbers,
13+ 23+ 33+ … + n3 is [ \frac{n(n + 1)}{2} ]2.