Question

Given that t_{7}=x+56,t_{8}=5x+41 and t_{9}=6x+44 are consecutive terms in an arithmetic sequence, find the following values:

x =
The difference between terms is:
The first term of this sequence is:

Answer 1

ok this is arithmetic sequence, do you remember the thing called the common difference?

Answer 2

mhm

Answer 3

so you know the difference of t8 and t7 will have the same difference as t9 and t8

Answer 4

you have an equation right there

Answer 5

replace the phrase will have the same with an equal sign

Answer 6

and you should be able to translate the difference between m and n as m-n

Answer 7

so it's like \[t_7=x+56\] = \[t_8 = 6x + 44\] ? i got lost

Answer 8

i want you to translate
"the difference of t8 and t7 is the same as the difference of t9 and t8"

First of all, the same as means equals so we can use the equal symbol =
so we now can write it as
"the difference of t8 and t7 = the difference of t9 and t8"
difference of m and n means m-n
so looking at the difference of t8 and t7 this means t8-t7
and looking at the difference of t9 and t8 this means t9-t8
now there should be an equal sign between these two expressions because that it what is says in the initial sentence above
so basically you have
t8-t7=t9-t8

Answer 9

replace t7,t8,and t9 will what they are in terms of x
then solve

Answer 10

don't forget parenthesis

Answer 11

if you want i will show you where you need them
(t8)-(t7)=(t9)-(t8)

Answer 12

so just insert those things in terms of x in there
and subtract now

Answer 13

then solve

Answer 14

oh, so i just plug the equations in according to the terms...hm

Answer 15

let me know what you get for x
or you can show your work so i can see where you went wrong in solve for x
or show me where you are getting lost

Answer 16

4635749.268

Answer 17

wowzers
that is definitely not correct

Answer 18

lol ik..

Answer 19

oh wtf i typed in the wrong thing

Answer 20

-3.78

Answer 21

ok we have the equation (t8)-(t7)=(t9)-(t8)
you were given: t_{7}=x+56,t_{8}=5x+41 and t_{9}=6x+44
so i'm going to replace t8 with 5x+41
and i'm going to replace t7 with x+56
and i'm going to replace t9 with 6x+44
like so
(t8)-(t7)=(t9)-(t8)
(5x+41)-(x+56)=(6x+44)-(5x+41)

Answer 22

did you get this far?

Answer 23

yes

Answer 24

so i typed the left side into the calc and got -12.14893616

Answer 25

ok distribute first if it helps you
like i'm talk about where you have negative in front of the parenthesis

Answer 26

and then i got 3.21276596 on the right side

Answer 27

for example i will do one side (only distributing)
5x+41-x-56=6x+44-(5x+41)
you try to take care of the distributing on the other side

Answer 28

6x + 44 - 5x - 41

Answer 29

great so we have the following equation
5x+41-x-56=6x+44-5x-41
now i'm going to regroup the terms on the left hand side so i have my like terms together on that particular side
like so
5x-x+41-56=6x+44-5x-41
you can regroup the terms on the other side

Answer 30

do you know what i mean?

Answer 31

4x - 13 = x + 3

Answer 32

is 41-56 equal to -13? or -15?

Answer 33

whoops -15

Answer 34

great so we have 4x-15=x+3

Answer 35

now we want to get our x terms on one side
and our constant terms on the other since we are trying to in fact isolate x

Answer 36

ah crap, that's the part im always iffy about

Answer 37

ok i will begin the process
we have 4x-15=x+3
I'm going to go ahead and subtract x on both sides giving me
(4x-15)-x=(x+3)-x
I did this because that means that there will be no x on one side
4x-15-x=x+3-x
4x-x-15=x-x+3
4x-x-15=3

so we have now


4x-x-15=3

Answer 38

i'm sure you can do the 4x-x part, right?

Answer 39

4x - 15 = 3
4x-15+15 = 3 + 15
4x = 18
x = 4.5

Answer 40

4x-x is not 4x

Answer 41

let me rewrite 4x-x
4x-x
is the same as saying 4x-1x

Answer 42

and 4-1 is not 4
but 3

Answer 43

eek i was thinking 5

Answer 44

so you should get a different x now

Answer 45

x = 3.6

Answer 46

hmmm
ok 3x-15=3 is right to say
then you added 15 on both sides
3x-15+15=3+15
3x+0=3+15
3x=18
so you are saying x is ?

Answer 47

6...lol omg

Answer 48

yes
so we have answered the first question

Answer 49

now the second question is asking for the common difference

Answer 50

just plug it in

Answer 51

and then count up the differences, im assuming

Answer 52

so it wants to know what t8-t7 equals


(note: this number will be the exact number as doing t9-t8 because this is a common difference number)

Answer 53

so d = 9

Answer 54

great

Answer 55

ok now we need to find the 1st term

Answer 56

a + d (n-1)

Answer 57

you know exactly what formula we will be using
we know that
\[t_n=t_1+d \times (n-1)\]

we know the 7th, 8th, and 9th term so we can use any one of them

so if we choose to use the 7th terms then we let n=7
and solve for the value that is t1

Answer 58

wherever you see an n replace it with 7

Answer 59

you will notice you will need to know what t7 is equal to

Answer 60

54 as t_7?

Answer 61

so x=6 and t7=x+56
so t7 cannot be 54

Answer 62

oh psht 62 xD

Answer 63

yeppie
ok we had the equation
\[t_n=t_1+d \times (n-1)\]
and you replace all the n's with 7 (you could have used even 8 or 9 since knew the values for also t8 and t9)

but anyways replacing n's with 7's we have
\[t_7=t_1+d \times (7-1)\]
you said t7 was 62
so we have
\[62=t_1+d \times (7-1)\]
also d represents the common difference which was 9
so we can also input that
\[62=t_1+9 \times (7-1)\]

Answer 64

this is your last equation to solve

Answer 65

i recommend doing order of operations on the left hand side where you can first

Answer 66

62=54

Answer 67

where is t1?

Answer 68

i think you meant 62=t1+54

Answer 69

one more step to go
you can do this
you need to find out what the first term in the sequence is
we need to isolate t1 to find that

Answer 70

how do you get t1 by itself
there is a plus 54 so we need to _________________

Answer 71

8 = t1

Answer 72

yep

Answer 73

HAHA YEAH

Answer 74

you did very well
math is like a puzzle sometimes
try to enjoy them

Answer 75

omg thank you so much; this was pretty fun xD

Answer 76

i have to admit i enjoyed it too
i always enjoyed trying to figure out where things go when it comes to math

Answer 77

do you tutor outside of open study? you'd be a really great tutor to have tbh