Question

Algebra 2 stuff

Answer 1

@shamil98 Let's have fun?

Answer 2

Do you know summation notation? \[
\sum_{k=0}^{n} a_k
\]

Answer 3

No, not really. Explain?

Answer 4

Okay. Do you know what a function is?

Answer 5

Yes.

Answer 6

Okay do you know what a sequence is?

Answer 7

Nope.

Answer 8

A sequence is a type of function, but the input must be a natural number

Answer 9

Natural numbers are numbers \(1, 2, 3, 4,...\)

Answer 10

So for example, a function lets you do \(f(2.5)\), but a sequence won't allow \(2.5\) as an input.

Answer 11

Functions are generally written like \(f(x)\) where \(x\) is the input. Sequences are usually written as \(a_n\) where \(n\) is the input. So \(a_1\) is to \(f(1)\) as \(a_n\) is to \(f(x)\).

Answer 12

This is just a definition, not a complicated concept or anything. Sequence is just a name of a certain type of function.

Answer 13

Now \[
\sum a_n
\]just takes a sum of a sequence.

I'll start off with very simple example.

Answer 14

Oh, I've done some sequences before like finding the 10th sum of
\[a _{n}=n+5\]
or something

Answer 15

We could let \(a_n=n^2\)

Answer 16

\[
\sum_{n=1}^3 n^2 = (1)^2+(2)^2+(3)^2
\]

Answer 17

The top 3 and bottom n =1
they represent the terms in which you substitute n right?

Answer 18

At the bottom when you say \(n=1\) you are specifying that \(n\) is the input variable and where the sum starts. Now at the top the \(3\) is where it ends.

Answer 19

Got it .

Answer 20

There are certain sums which it is important to know.

Answer 21

\[
\sum_{i=1}^ni=\frac{n(n+1)}{2}
\]

Answer 22

I've seen that formula before, when I was studying for the SAT..

Answer 23

\[
\sum_{i=1}^nc=\underbrace{c+c+c+\ldots +c}_n = nc
\]

Answer 24

\[
\sum_{i=1}^nca_i = c\sum_{i=1}^na_i
\]You can pull out constants...
\[
\sum_{i=1}^n(a_i+b_i) = \sum_{i=1}^na_i+ \sum_{i=1}^nb_i
\]You can sum over particular terms.

Answer 25

In algebra 2 you learn about two special types of series:

Arithmetic series. They come in the form: \[
a_n=a_1+(n-1)d
\]Where \(d\) is the difference between any two consecutive terms.

Geometric series. \[
a_n=a_1r^{n-1}
\]Where \(r\) is the ratio between two consecutive terms.

Answer 26

I don't necessarily want you to memorize it. To understand it a bit and forget it later is fine. Knowing this, though, can help you help others on this site.

Answer 27

Thanks, I'll try and memorize it though, the more the knowledge the better :D

Answer 28

\[
\sum_{i=1}^n a_i=\frac 12n (a_n+a_1)
\]When \(a_i\) is an arithmetic series.


\[
\sum_{i=1}^na_i=a_1\frac{1-r^n}{1-r}
\]When \(a_i\) is a geometric series.

Answer 29

Oh, by the way "series" is just a word for a sum of a sequence.

Answer 30

This is a pretty quick overview of what goes on with sequences in algebra 2.

Answer 31

@shamil98

Would you like a test question?

Answer 32

Sure.

Answer 33

What is the sum of all numbers divisible by 8 from 1 to 1000.

Answer 34

hmm..

Answer 35

Our sequence is \[
8,16,24,\ldots,1000
\]

Answer 36

Arithmetic or geometric?

Answer 37

The first number greater than 1 divisible by 8 is 8,16,24,32,36 and so on..

Answer 38

numbers*

Answer 39

@shamil98 What is the formula for this sequence?

Answer 40

\(a_1=8\).\[
a_n=a_1+(n-1)d
\]You gotta find \(d\).

Answer 41

arithmetic btw

Answer 42

(You technically don't need the formula but for exercise let's get it).

Answer 43

gah, im confused ..
1000/8 = 125
so 1000 is the largest possible number that can be divided by 8..

Answer 44

and the smallest one is 8.

Answer 45

Use the formula wio gave you

Answer 46

an = 8 + (n-1)d

Answer 47

What's an? :D

Answer 48

in this case

Answer 49

@shamil98 Try using `\[a_n=...\]` \[
a_n=\ldots
\]

Answer 50

wait i know this..
s = 1/2 (1st term + last term) (last term)
right?..

Answer 51

You should really start of with questions like... determine whether the following sequences are arithmetic or geometric.

Answer 52

\[S_n=\frac 12 n (a_1+a_n)
\]But I already wrote that above.

Answer 53

We don't need to give shamil kiddy questions

Answer 54

Eh best to start of with the basics then get into all the notation etc.

Answer 55

s = 124/2 (8+1000)
s = 62496?..

Answer 56

Why 124?

Answer 57

8*1 = 8
8*125 = 1000
125 - 1 = 124
am i wrong?..

Answer 58

There are 125 numbers between 1 and 125.

Answer 59

By the way, the explicit formula in this case is: \[
a_n=8+(n-1)8 = 8n
\]

Answer 60

a_n = 8 + (125-1)(8)
then?.

Answer 61

Wio, am i right or wrong bout the sum tho? o.o

Answer 62

\[
a_{125}=8+((125)-1)(8)
\]

Answer 63

Since \(a_n\) is a sequence of \(n\), we don't need to substitute \(125\) into it.

Answer 64

You would be correct about the sum if you used \(n=125\)

Answer 65

so,
s = 125/2(8+1000)
s= 63000

Answer 66

That is right.

Answer 67

Okay, want a quick and easy geometric one?

Answer 68

Sure

Answer 69

A man is given 1 grain of rice on the first day. Each day he is given twice the number he was given the previous day. How many grains is he given by the 64th day?

Answer 70

1 = 1
2 =2
3 = 4
4= 8
5 = 16
and so on..

Answer 71

Yes.

Answer 72

S = (first term)(1-r^n)/(1-r)

Answer 73

Although I am bit curious what would r represent in this formula?

Answer 74

ratio between two consecutive terms.

Answer 75

so the ratio in this case is 1/2 right?

Answer 76

No, it is 2

Answer 77

oh.

Answer 78

2^n -1

Answer 79

n represents the number of days ? correct?..

Answer 80

yeah

Answer 81

so
2^64 - 1?

Answer 82

18446744073709551615
is your answer o.o

Answer 83

http://www.wolframalpha.com/input/?i=%281-2%5E%2864%29%29%2F%281-2%29

Answer 84

yeah that looks right.

Answer 85

nice number looool

Answer 86

How many pounds of rice?

Answer 87

18446744073709551615 pounds of rice :D

Answer 88

that's a lot of rice man.

Answer 89

Integrals are a sumation.

Answer 90

\[
\lim_{\Delta x\to 0}\sum_{i}f(x_i)\Delta x = \int f(x)\;dx
\]