Question

What is the d value in the following arithmetic sequence? -7, -2, 3, 8, …

Answer 1

what does the value of d do for us in an arithmetic sequence?

Answer 2

it's the common difference but that doesn't really make sense to me.. the course explains it as the common number but what does that mean?

Answer 3

good, it means that in order to get from one number, to the next number, we add the same amount each time. in this case:

-7 + d = -2
-2 + d = 3
3 + d = 8

Answer 4

Do you need some explanation of this in general, like what this all about, `difference` `sequence` `arithmetic` and all this ?

Answer 5

ohhh wait so you would be adding 5each time. is that the value of d?

Answer 6

you would be adding 5 each time :) so yes, d=5

Answer 7

okay thank you very much

Answer 8

youre welcome

Answer 9

aswwsa, can you find the 15 number in this sequence ?

Answer 10

I'll post a brief explanation of other notations (of how this is usually written) okay ?

Answer 11

that would help a lot

Answer 12

Lets take your sequence for example \(\normalsize\color{black}{ -7, ~-2, ~3, ~8}\)

the first term is `-7`
the first term is `-2`
the first term is `3`
the first term is `8`

In general though, the terms are labeled like this:

\(\large\color{black}{ a_1}\) first term

\(\large\color{black}{ a_2}\) second term

\(\large\color{black}{ a_3}\) third term

\(\large\color{black}{ a_4}\) forth term


and on.....

Answer 13

So in any sequence, how would you write the 100th term ?

Answer 14

that would be al00 right?

Answer 15

yes

Answer 16

Would it make sense to say that to get from \(\large\color{black}{ a_1}\) to \(\large\color{black}{ a_{100} }\) you add the difference (whatever it is) 99 times ?

Answer 17

you mean like add d 99 times? no

Answer 18

YES !!!!very good :)

Answer 19

So we can say that

\(\large\color{black}{ a_{n}=a_1+d(n-1) }\) would this make sese ?

Answer 20

*sense

Answer 21

ohh yeah that makes more sense so say I'm finding the 25th term you would it end up like this.. an=-7+5(25-1) ?

Answer 22

yes if it is \(\large\color{black}{ a_{25}=-7+5(25-1) }\) if you were to look for a 25th term in our sequence.

and what would \(\large\color{black}{ a_{25} }\) be equal to, can you calculate that ?

Answer 23

(Sorry for late reply, just finished my college application)

Answer 24

it's fine. it would be 113?

Answer 25

yes !

Answer 26

Can you find the 50th term ?

Answer 27

that would be 238

Answer 28

yes !

Answer 29

Now, lets say if you needed to add all 50 terms up, you would use a formula


\(\LARGE\color{midnightblue}{ \rm S_n=\color{red}{\frac{1}{2}(a_1+a_n)}× n }\)

Sn is 'sum of n number of terms'
n is number of terms

Answer 30

\(\LARGE\color{midnightblue}{ \rm S_{50}=\color{red}{\frac{1}{2}(-7+238)}× 50 }\)

Answer 31

ok 1 more quick question before that.. how would you find the 4 arithmetic terms between -21 and -36

Answer 32

4 arithmetic means between 21 and 36

Answer 33

-21 and -36
••••

So, just for imagination will say

-21, __ , __ , __ , __, -36

\(\normalsize\color{blue}{ -36=-21+d(5)}\) (because you should add d 5 times to get from -21 to -36)

Answer 34

Solve for d, and then you will be able to fill in the missing terms.

Answer 35

if you had N missing means, you would add d `N+1` times

Answer 36

it would be -3 I think?

Answer 37

Yes

Answer 38

adding 21 to both sides, ad then dividing both sides by -3 :)

Answer 39

So if d=-3

then the 4 missing terms are ?

Answer 40

it'll be -24, -27, -30, -33

Answer 41

yes !

Answer 42

\(\LARGE\color{blue}{ S_n=\underline{\frac{1}{2} (a_1+a_n)} ×n }\)


Sn is `sum of n number of terms`
n is ` number of terms `

I underlined the average term, makes sense why this is the average term in ANY sequence ?

Answer 43

yes I think that makes sense. another question.. sorry I have so many.. how would you find S25 for -7+ -2 + 3 + 8 + ... ?

Answer 44

Use the formula I posted

Answer 45

ohhh would you just find the value of the 25th term and add that to the first the take half of that? what would you multiply it by then, what is n?

Answer 46

a25 = a1 + d(n-1) → a25=-7+5(25-1)

Answer 47

a25 = ?

Answer 48

113

Answer 49

yes

Answer 50

Now use the S(n) formula

Answer 51

S(n)=½(a1+an)×n

Answer 52

S(25) = ½ (-7 + 113) × 25

Answer 53

so it would be 1325?

Answer 54

Yes, very good !

Answer 55

You are a pleasure to work with :)

Answer 56

thank you so much for all you help

Answer 57

all of your help* oops

Answer 58

You welcome, have fun !