Question

A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying Sk+1 completely.

Answer 1

Sn: 1 + 4 + 7 + . . . + (3n - 2) = n(3n - 1)/2

Answer 2

What do you need help with?

Answer 3

i.e. which part is confusing you?

Answer 4

figuring this out?

Answer 5

all of it, it confuses me can you do step by step with me?

Answer 6

No prob.

Answer 7

Does the question say anything about "Induction"?

Answer 8

The first part is easy: just look at the "Sn" part and replace the "n"s with "k"s.

Answer 9

with me?

Answer 10

okay with you

Answer 11

So you're there?

Answer 12

yes

Answer 13

Because you seem to be, um, preoccupied...

Answer 14

no im here sorry i was just messaging my teacher quick

Answer 15

Ok, so you're ready?

Answer 16

yes

Answer 17

You sure

Answer 18

yes positive

Answer 19

Ok. First, explain to me why you're confused.

Also, explain to me what you think the question is saying and what it's asking you... Then I'll walk you through step by step.

Answer 20

im confused because im not sure exactly how to break it down . & I think its asking me to simplify Sk & Sk+1

Answer 21

First, notice how the series starts at 1 and then each term that follows is equal the previous term plus 3.
1st term = 1
2nd term = 1+3 = 4
3rd term = 4+3 = 7
4th term = 7+3 = 10
...
Right?

Answer 22

okay im following

Answer 23

Also notice how the "k-th" term in the series:
kth-term = 3k -2

where "k" can be any positive whole number.
Thus, for the 5th term, we can either calculate it as:
5th-term = 4th-term +3 = 10+3 = 13
Or, we can calculate it as:
5th-term = 3(5) -2 = 13

Either way, we get the same result....

Answer 24

okay!

Answer 25

So you get what I'm saying?

Answer 26

yea im on your page

Answer 27

So that's what the left side of your equation was saying: Take the first "n" numbers of that form and add them all together...

Answer 28

okay!

Answer 29

By the way, in your picture, what's that thing behind you? For a while I thought it was a sock puppet you were holding (weird!). Anyway, I digress...

Answer 30

Its my feet !!

Answer 31

No, the black thing--I swear it looked to me like your shoulder was really your forearm holding up a black hand puppet (Note: I might be crazy).

Answer 32

its a coat on a chair lol

Answer 33

Ok, so anyway, if I asked you to sum up the first 7 numbers of that form, you could easily do it on your calculator, right? You would just add:

1+4+7+10+13+16+19 = 70

Answer 34

Easy, right?

Answer 35

But what if I asked you to add up the first 1000 numbers of that form--that would be a real pain in the neck. And that's where the right-side comes in.

Answer 36

the first part was easy lol !

Answer 37

What do you mean?

Answer 38

meaning adding the first 7 numbers , but 1000 would be a pain

Answer 39

So the right side tells us that the sum of the first n numbers of that form is equal to:

\[\frac{ n(3n-1) }{ 2 }\]

So, when n=7, we get
7(20)/2 = 70
(the same thing we got when we added things up individually)
and we can do that (much simpler) calculation for any "n" we want.

Answer 40

Got it?

Answer 41

got it. so thats it?

Answer 42

What do you mean?

Answer 43

was that all i had to do?

Answer 44

Yeah, the last part of the question is easy: you just replace the "n"s with "k"s and then, for the second part, just replace the n's with "k+1"s and simplify.

The question itself is easy, but that's because its trying to slowly introduce something that's more complicated (so it's good that you understand how it all works now).

Answer 45

Also, something you'll see pretty soon is called "Sigma notation"--it's a way of concisely representing that sum on the lefthand side. It looks like this:

\[S _{n}=\sum_{k=1}^{n}(3k-2)\]

What it's saying is "start at k=1 and add all the terms from 'k' up to 'n'"

Answer 46

so Sk: (3k - 2) = k( 3k -1) /2
or do i plot 7 in for k now and then in sk+1 simplify it all?

Answer 47

No, because

\[S _{k} = 1 + 4 + 7 + 10 + .... + 3k-2\]
(i.e. not just "3k-2" but all the terms from 1 to "3k-2", counting by threes)

And that sum is equal to
\[\ S _{k} = \frac{ k(3k-1) }{ 2 }\]

Answer 48

so do i put exactally what you put for sk or do i put what we did individually

Answer 49

what do you mean?

Answer 50

meaning for sk= do i put 1+ 4 + 7 _ 10+... + 3k -2
or do i put 1st term = 1
2nd term = 1+3 = 4
3rd term = 4+3 = 7
4th term = 7+3 = 10

Answer 51

You write it the first way ( 1+4+7+...+3k-2)
The second way was just me showing you where each term came from and what that whole Sn junk was all about...

Answer 52

okay dokie

Answer 53

and remember, you want
1+4+...+3k-2 = k(3k-1)/2
ie. Don't forget the second half!

Answer 54

and for k+1-th term, it's just
1+4+...+3(k+1)-2, which simplifies to
...+3k+1

etc.

Answer 55

Sk+1: 1+4+7+...+3(k+1)-2
= + 3k+1

Thatd be the second half right?

Answer 56

You mean for S_k+1?

Answer 57

that would be the lefthand side of the second half

Answer 58

DOn't forget the righthand side

Answer 59

1+4+...+3k+1

Answer 60

okay

Answer 61

Then the right hand side being "(k+1)(3(k+1)-1)/2" and then simplify

Answer 62

Whew! I'm exhausted!! I hope you appreciate this!! :)