Sequences An infinite sequence is a function whose domain is the set of positive integers. The function values a 1, a 2, a 3, a 4, …..a n,… Are the terms of the sequence. If the domain of a function consists of the first n positive integers only, the sequence is a finite sequence.
Factorial If n is a positive integer, n factorial is defined as n! = 1 · 2 · 3 ·4 · · · ·(n-1) · n. As a special case, zero factorial is defined as 0! = 1
Definition of Summation Notation The sum of the first n terms of a sequence is represented by ∑ a i = a 1 + a 2 + a 3 + a 4 + · · · + a n Where I is called the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation. n i= 1
Properties of Sums 1.∑ c = cn, c is a constant 2.∑ca i = c ∑ a i, c is a constant 3.∑(a i + b i ) = ∑ a i +∑ b i 4.∑ (a i - b i ) = ∑ a i -∑ b i Note* Summations here go from i = 1 to n
The sum of the terms of a finite or infinite sequence Series
The sum of the first n terms of the sequence is called a finite series or the partial sum of the sequence and is denoted by a 1 +a 2 +a 3 +a 4 +…+ a n = ∑ a i (summation goes from 1 to n) The sum of all the terms of the infinite sequence is called an infinite series and is denoted by a 1 +a 2 +a 3 +a 4 +…+ a i +…= ∑ a i (summation goes from 1 to infinity)
Arithmetic sequence A sequence is called arithmetic if the differences between consecutive terms are the same. So the sequence a 1, a 2, a 3, a 4, …, a n, … Is arithmetic if there is a number d such that a 2 – a 1 = a 3 – a 2 = a 4 – a 3 = … = d The number d is the common difference of the arithmetic sequence.
Form for the nth term of an Arithmetic Sequence The nth term of an arithmetic sequence has the form a n = dn + c Where d is the common difference between consecutive terms of the sequence and c = a 1 - d
Sum of a Finite Arithmetic Sequence The sum of a finite arithmetic sequence with n terms is given by S n = (n/2) (a n + a n )
The sum of the first n terms of an infinite sequence is called the nth partial sum
Geometric sequence A sequence is a geometric sequence if the ratios of consecutive terms are the same. So the sequence a 1, a 2, a 3, a 4, …, a n, … is geometric if there is a number r such that a 2 / a 1 = a 3 / a 2 = a 4 / a 3 = … = r ≠ 0 The number r is the common ratio of the sequence.
Nth term of a geometric sequence The nth term of a geometric sequence has the form a n = a 1 r n-1 Where r is the common ratio of the sequence.
Sum of a finite Geometric sequence The sum of a finite geometric sequence, a 1,a 1 r,a 1 r 2, a 1 r 3,a 1 r 4, …, a 1 r n-1 With common ratio r ≠1 is given by S n = ∑ a 1 r i-1 = a 1 [(1- r n )/(1-r)] (summation goes from 1 to n)
The sum of an infinite Geometric Series If |r| < 1, then the infinite geometric series a 1,a 1 r,a 1 r 2, a 1 r 3, …, a 1 r n-1, … has the sum S = ∑ a 1 r i = a 1 / (1-r)