Sequence – An ordered list of numbers. (ie. 3, 6, 9, L2, 14, …… or 4,8, 16,32, ….). Lorem Ipsum is simply dummy text of the printing and typesetting industry. Lorem Ipsum has been the industry’s standard dummy text ever since the 1500s, when an unknown printer took a galley of type and scrambled it to make a type specimen book. It has survived not only five centuries, but also the leap into electronic typesetting, remaining essentially unchanged. It was popularised in the 1960s with the release of Letraset sheets containing Lorem Ipsum passages, and more recently with desktop publishing software like Aldus PageMaker including versions of Lorem Ipsum.
Term – Each element of a sequence. The first term is referred to as u1 and the second term is u2 the third term is
u3, and so on. The nth term is referred to as un.
Sequences may be finite (end at a certain term value) or infinite (continues to infinity) and denoted by three
dots at the end of the number sequence.
Sometimes a pattern can lead to a general rule for finding the terms of a sequence. The rule is called the
formula for the nth term, un.
Arithmetic Sequence – a sequence that has the same common difference between any pair of consecutive terms.
A common difference is obtained by subtracting a term from the previous term. lf this value is, constant for all terms in the sequence then the sequence is arithmetic.
i.e: 5,8, 11,14, 17, … common difference = 3
General Term of an Arithmetic Sequence
un = u1+ (n – l)d
un = nth term or general term
un = first term
n = number of terms
d = common difference
Geometric Sequence – a sequence that has the same common ratio between any pair of consecutive terms. A common ratio is obtained by dividing any term by the preceding term. lf this value is
constant for all terms in the sequence then the sequence is geometric
i.e: 2,6,18, 54,…. common ratio = 3
General Term of an Geometric Sequence
un = u1nn-1
un = nth term or general term
un = first term
n = number of terms
r = common ratio
Sequences and summations
The sum of the, first n terms of a sequence is
represented by summation notation.
S1000 = 1 + 2 + 3 + …..+ 999 + 1000
S1000 = 1000 + 999 + 998 + …..+ 1 + 2Example :
2S1000 = 1001 + 1001 + 1001 …..1001 + 1001 + 1001
2S1000 = 1001(1001)
S1000 = \(\frac{1001(1000)}{2}\)= 500500
Example: An empty barrel can hold L000 L of water. Alison pours 500 L of water into the barrel, then Jack pours 250 L into the barrel, then Marta pours 125 L into the barrel. Suppose the process goes on forever. Will the barrel ever fill? Will the barrel overflow?
SolutionThe amount may be written as a gyrometric series
500 + 250 + 125 ………..
The more term are add to the series. The close we get to 1000 L. we say limit of the infinite geometric series is 1000 L
Sum of an Infinite Geometric Series
S∞= u 1 + u 1r2 + u 1r3……..
S∞= \(\frac{u_1}{1-r}\) ,-1<r<1,(r<1)