For each formula, describe the four quantities. Then explain how you can find the fourth quantity if you know the values of the other three. A) an=a1 + (n-1) d B) sn= n/2 ( a1-an) C) an=a1r ^ n-1 D) Sn = a1 ( 1-r^n) / 1-r

Question

For each formula, describe the four quantities. Then explain how you can find the fourth quantity if you know the values of the other three.  A) an=a1 + (n-1) d  B) sn= n/2 ( a1-an)  C) an=a1r ^ n-1  D) Sn = a1 ( 1-r^n) / 1-r

anonymous · Answer

In an Arithmetic Sequence the difference between one term and the next is a constant. A1 is the first term, d is the common difference between each and every number after the first term and n is the number of terms in which one would plug in 4. 
a_n  = a_1  + (n – 1)d  
This is the formula of the arithmetic sequence, in an arithmetic sequence it is going from one number by a common difference, when we sum  up a number of terms, we first take the number of terms multiplied by the first term + the last term divided by 2. 
S_n=n(a_1+a_n )/2
In a geometric series, there is a series where each successive term is the previous term multiplied by a fixed value, common ratio (r). 〖a_1  r〗^(1-1),〖a_1 r〗^(2-1),〖a_1 r〗^(3-1),〖a_1 r〗^(4-1) 
a_n=a_1 r^(n-1)
We can then find the sum of the geometric series using this formula. Which is done by distributing the term “a” to 1 minus the common ratio with a power containing the number of terms divided by 1 minus the common ratio. 
S_n=a(1-r^n )/((1-r))

anonymous · Answer

I'm just wondering if I described the four quantities right and if I'm supposed to solve for n only.

anonymous · Answer

@zepdrix , @mathstudent55 , @ganeshie8

ganeshie8 · Answer

Overall it looks good, I see one mistake :
whats the difference between a `sequence` and a `series` ?

ganeshie8 · Answer

You're using both the terms as synonyms... but they mean different things... do you know hte difference between the two ? :)

anonymous · Answer

a series is a sum of terms and a sequence is a list of numbers with some form of pattern.

anonymous · Answer

But may I know where I made the small mistake?

anonymous · Answer

?

anonymous · Answer

@ganeshie8 , would you be able to show me where you had spotted the mistake?

ganeshie8 · Answer

exactly !

below is a sequence :
$$\large  1,~2,~3,~4,~\cdots$$

below is a series:
$$\large  1+2+3+4+\cdots$$

ganeshie8 · Answer

A) an=a1 + (n-1) d `arith metic sequence formula`  
B) sn= n/2 ( a1-an)  `arithmetic series formula`
C) an=a1r ^ n-1   `geometric sequence formula`
D) Sn = a1 ( 1-r^n) / 1-r  `geometric series formula`

ganeshie8 · Answer

`Sn`  is a series formula,
`an` is a sequence formula...  okay ?

anonymous · Answer

okay I will make sure to do that and it says "find the fourth quantity", but im not quite sure what its talking about.

ganeshie8 · Answer

look at the arithmetic sequence formula :
$$\large a_n  = a_1  + (n – 1)d  $$

how many variables(quantities) do you see ?

anonymous · Answer

4

ganeshie8 · Answer

and suppose if i tell you ANY 3 of those variables, 
will you be able to solve for the other variable ?

anonymous · Answer

an, a1, n, and d

ganeshie8 · Answer

yes!

anonymous · Answer

yes

ganeshie8 · Answer

good :) tell me how you will solve the other variable ?

ganeshie8 · Answer

just describe,
this is more of a writing problem...

anonymous · Answer

so if i were to solve for a1 I would first divide dn on both sides and add d on both sides, which would leave me with a1=d+ n(a-d)

ganeshie8 · Answer

kindof.. 

$$\large a_n  = a_1  + (n – 1)d$$

to solve $a_1$, you would simply subtract $(n-1)d$ both sides :

$$\large a_n  - (n – 1)d = a_1  $$

anonymous · Answer

so i don't distribute then, i see.

ganeshie8 · Answer

yees we don't need to distribute cuz $a_1$ was isolated already...

anonymous · Answer

right

ganeshie8 · Answer

just add this step to the other 3 formulas as well, good luck !

anonymous · Answer

alright thanks for the help