Question

I WILL FAN AND MEDAL
2 SERIES AND SEQUENCE QUESTIONS!!

Answer 1

1. What is the hundredth term of the geometric sequence having r=1/2 and u1=-1/2?

Answer 2

Well, \(r=-1/2\) means that you are multiplying times (-1/2) to obtain every succeeding term.

Answer 3

Yes okay

Answer 4

So, from \(u_1\), you will multiply times (-1/2) exactly 99 times, to obtain \(u_{100}\).

Answer 5

-1.577*10^-30

Answer 6

So, \(u_{100}=u_1\times \left(-\frac{1}{2}\right)^{99}\)

Answer 7

oh sorry, \(r=1/2\).

Answer 8

So, by the same logic you would be getting \(u_{100}=u_1\times \left(\frac{1}{2}\right)^{99}\)

Answer 9

okay

Answer 10

then knowing \(u_1=1/2\) you have a nice way of re-writing it.

Answer 11

the most similar multiple choice option I have is: -1/2^100 . could that be correct?

Answer 12

Everything "could" be correct ... can you show me how you got it (to avoid guessing) ?

Answer 13

um.. I just took what you said about how the 1/2 is applied 99 times and saw that the only answer choice like that is -1/2^100

Answer 14

Well, what I was interested in hearing is actually the following:

\(u_{100}=u_1\times \left(r\right)^{99}\) (by definition)
\(u_{100}=(-1/2)\times \left(\frac{1}{2}\right)^{99}\) (Substitution)
\(u_{100}=-\left(\frac{1}{2}\right)^{99+1}=\) ....

Answer 15

of course, what you said

Answer 16

haha I'm still learnign, but I understand the jist.

Answer 17

2. The first term of an arithmetic sequence is -15 and the fifth term is 13. Find the fortieth term.

Answer 18

Well, the terms in an arithmetic sequence always abide by the formula
\(\color{black}{ \displaystyle u_n=u_1+d(n-1) }\)

Answer 19

where
d - the common difference between any nth and (n-1)th term.
\(u_n\) is the nth term.

Answer 20

okay

Answer 21

Given this formula, how would you express the 5th term \(u_5\) ?

Answer 22

un=u1+-28(13-1) ? that is prob totally wrong

Answer 23

yes, it's wrong :(

Answer 24

whoops

Answer 25

Well, we said that nth term would be modeled as follows:
\(\color{black}{ \displaystyle u_n=u_1+d(n-1) }\) right?

Answer 26

So, if I wanted to express the 8th term this way, I would say
\(\color{black}{ \displaystyle u_8=u_1+d(8-1) }\)
\(\color{black}{ \displaystyle u_8=u_1+7d }\) <--- Correct?

Answer 27

oh okay yes

Answer 28

So how would you do the 5th term?

Answer 29

u5=u1+d(5-1)
u5=u1+4d

Answer 30

yes, exactly!

Answer 31

\(\color{black}{ \displaystyle u_5=u_1+4d }\)

Answer 32

You were given that "the first term of an arithmetic sequence is -15 and the fifth term is 13".

Alternatively speaking you know that
(1) \(\color{black}{ \displaystyle u_5=13 }\)
(2) \(\color{black}{ \displaystyle u_1=-15 }\)

So, in order to find the common difference \(d\) between the terms (the number that you add to obtain each succeeding term), you can just plug (1) and (2) into \(\color{black}{ \displaystyle u_5=u_1+4d }\) to solve for \(d\).

Answer 33

13=-15+4d ?

Answer 34

Yes, exactly !

Answer 35

now go on to solve for d:)

Answer 36

d=7

Answer 37

Perfect!

Answer 38

So, we know that d=7.
(In other words, you add 7 to say 100th term to obtain 101st ... Yes?)

Answer 39

yes

Answer 40

Now, the same way you expressed the 5th term, please go ahead and write the expression for the 40th term \(u_{40}\).

Answer 41

Wait, I think I got it. i looked ahead and got the answer to be: 258.. is that right?

Answer 42

i guess i just needed help finding d

Answer 43

At first, we just found d=7, didn't we?
At second, yes, your answer is correct.

Answer 44

Thank you so much! I have to go, i really apreciate your help

Answer 45

\(\color{black}{ \displaystyle u_{40}=u_1+d(40-1) }\)
\(\color{black}{ \displaystyle u_{40}=u_1+39d }\)

we know
\(u_1=-15\)
\(d=7\). Consequentially,

\(\color{black}{ \displaystyle u_{40}=-15+39\times 7=258 }\)

Answer 46

I also have to go:) )Timely)
YW