Question

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Answer 1

Did you find the common difference and final term yet?

Answer 2

no ::/ im really confused.

Answer 3

The common difference is the distance between each successive term.
\[\large d=a_{n+1}-a_n\]

Answer 4

ok

Answer 5

Try (-7/12) - (-1/4)
and ( -11/12) - (-7/12) to see that the difference is indeed the same.

Answer 6

ok

Answer 7

-0.33

Answer 8

?

Answer 9

Right, though I recommend using the exact -1/3 instead of the rounded approximation of -0.33

Answer 10

ok

Answer 11

Groovy. Alright, now to find the fortieth term, think of how many times you need to add -1/3 to -1/4 to get the fortieth term.

Answer 12

39 times

Answer 13

Exactly! So, do that and find out what the 40th term is.

Answer 14

ok

Answer 15

can i just square to the 39th?

Answer 16

No, but you can multiply. Adding the same number 39 times is the same as that number *times* 39.
i.e. (-1/3)×39

Answer 17

that would be -13 ?

Answer 18

Right, now that is what was added to your first term of -1/4, so the last term is -53/4.

Answer 19

ok

Answer 20

Alright, now we have all the pieces necessary to plug into the formula.
Do you know the formula for the sum of an arithmetic series?

Answer 21

It has to do with S(n) ..right?

Answer 22

Yes, it looks like S(n) = . . . .

Answer 23

If you don't remember it, there's an easy way to figure it out.

Answer 24

ok

Answer 25

(I never bother memorizing the formula, I just reason my way through it..)

Answer 26

Consider this other example:
A set of four numbers {1, 5, 9, 13}, which has a common difference of 4.
If you add the first and last terms, you get 14, but you also get 14 if you add the second and the second-to-the-last terms (5+9), right?
So there are 2 pairs of numbers that add up to the same thing as the first and last terms added together.

So, in general, the sum of an arithmetic series is (n/2) pairs of numbers that are all equal to (1st term + last term).

Answer 27

ok i remember the formula:
Sn= n/2[ (2a^1 + (n-1)d ]

Answer 28

So, that is a quick way of reasoning through to derive the following formula:
\[\large S(n)=\frac{n}{2} \times (a_1 + a_n)\]

Answer 29

oh ok

Answer 30

so -1/3 goes for n?

Answer 31

No, n is the number of terms (40)

Answer 32

ok

Answer 33

what about the a?

Answer 34

The a's are the terms. a_1 is the first term, a_2 is the second term, . . .a_n is the last term.

Answer 35

ok

Answer 36

S(n) = 40/2 (-1/4 + -53/4) = -270 ? :)

Answer 37

That looks correct to me.

Answer 38

thank you so much for explaining this to me :)

Answer 39

You can also try the other formula to verify that it works
\[\large S(40)=\frac{n}{2}(2(-1/4)+(40-1)(-1/3))\]

Answer 40

Sorry, meant to replace that (n/2) with (40/2)...

Answer 41

You're welcome. Think you got a better handle on this stuff now?

Answer 42

Yes I have better understanding of it now.