Question

What is the difference between a sequence and series?

Describe the common difference for an arithmetic sequence in which the terms are decreasing?

Answer 1

A sequence is an ordered set of "things".
\[\huge\ {a,b,c,d,e,f,g,...} \]

A series is a summation over a sequence.
\[\huge \sum_{}^{}({a,b,c,d,e,f,g,...}) = a + b + c + d + e + f + g + ... \]

Answer 2

I never quite saw the point of defining "sequence" or "set" differently. To be more rigorous, though, a sequence is essentially a set with (presumably) a pattern to each further unit:\[\{x_1,x_2,x_3,\cdots,x_{n-1},x_n\}\mid x_i=f(x_{i-1}),\forall i\in\mathbb N\]

A series is a summation of the sort:\[\sum_{i=a}^bf(x_i),\forall(b\geq a,i\in\mathbb N)\]

Answer 3

A geometric sequence is a sequence of the form
\[\huge\ a,ar, ar^2, ar^3 \]
where 'a' is the starting term and 'r' is the common ratio. If 'a' is positive, then the value of 'r' will determine whether the terms will increase or decrease. If 'r' is greater than 1 (ie r>1)
then the terms will increase, while on the other hand, if 'r' is less than 1 r<1
then the terms will decrease. Finally, if 'r' is equal to 1, then the terms will remain the same.
If r>1 then we can write 'r' as r=1+k

Answer 4

ok i understand thanks!

Answer 5

please give him a easy example.
http://www.mathsisfun.com/algebra/sequences-series.html

Answer 6

and then whats the common difference for an arithmetic sequence in which the terms are decreasing?

Answer 7

common deffrence means 2nd term-1st term

Answer 8

is the common difference negative?

Answer 9

an airthmetic sequence 1,2,3,4........
then
d=2nd term-1st term
d=2-1
d=1

Answer 10

it can be negative or postive. If sequence is decreasing then it will be negative.

Answer 11

A sequence is a list of numbers. The order in which the numbers are
listed is important, so for instance

1, 2, 3, 4, 5, ...

is one sequence, and

2, 1, 4, 3, 6, 5, ...

is an entirely different sequence.

Answer 12

@kenneyfamily do you understand?

Answer 13

yes thank you!

Answer 14

Most of it has been covered, but I also want to add that there is another way of thinking about series that becomes very useful. We can think of a series as a sequence itself. It is a sequence of what we call "partial sums." So, if we have the sequence 1,2,3,4,5,..., and we talk about the series that sums every term of that sequence, there is the traditional summation way of thinking about it: \(\Sigma_{i=1}^\infty i\), but you can also think of it as a sequence of partial sums: 1, 1+2, 1+2+3, 1+2+3+4, ... This is useful when talking about convergence and other topics.

Answer 15

thanks @ganeshie8