Question

Could anyone help me? I'm new here so I'm not sure how this works... But here's my question:

Find the sum of the geometric sequence.

1, 1/4, 1/16, 1/64, 1/256

I know I need to use sn= n/2 (a1+an) formula and then I'd get this 5/2(1+1/256) but I'm having trouble going from here...

Answer 1

@mathmale Can help you!!:) He's really smart!!!! He helped me with most of my work!!!:)

Answer 2

ok thank you :)

Answer 3

Let's jump right in. The first term of this geometric sequence is what? a = ?
The common ratio is what? r = ?

If we wanted to find the sum of the infinite series, that sum would be \[S=\frac{ a }{1-r }\]

Answer 4

But here we want only the sum of the first 5 terms of the sequence, right?

Answer 5

Yup

Answer 6

You were using the formula sn= n/2 (a1+an). Let's see how that works out. Your a1 is just 1, and n is 5. You got 5/2(1+1/256. Without checking this formula, let's try evaluating it.

Answer 7

that's where I got stuck

Answer 8

I see the following:\[\frac{ n }{ 2(a _{1}+a _{n}) }\rightarrow \frac{ 5 }{ 2(1+\frac{ 1 }{ 256 }) }\]

Answer 9

Let's simplify the denominator. Focusing on 1+1/256, recognize that the LCD here is 256.
then 1+1/256= (256+1)/256, or 257/256.\[\frac{ 5 }{ 2(\frac{ 257 }{ 256 }) }\]

Answer 10

Can you simplify this? Hint: Don't do any multiplication. Instead, reduce!

Answer 11

Let me try, one sec.

Answer 12

I got 640/257

Answer 13

Same here. Now we need to add up those 5 actual terms and see whether they do sum up to 640/257. Do you have a TI calculator handy?

Answer 14

I do, but the thing is that that answer isn't one of my answer choices

Answer 15

the actual formula is
\(\large \dfrac{n}{2}(a_1+a_n)\)

Answer 16

these are my options:
341

1/292

1/768

341/ 256

Answer 17

Unfortunately, that means we have a mistake somewhere. Note that hartnn has provided the correct formula. Would you mind trying that formula instead?

Answer 18

hey wait!

Answer 19

thats the sum formula for ARITHMETIC sequence :P
not geometric

Answer 20

Unfortunately, that means we have a mistake somewhere. Note that hartnn has provided us with an alternative formula. Would you mind trying that formula instead?

But I have my doubts! hartnn's formula is for the sum of an ARITHMETIC sequence, whereas ours is a GEOM sequence.

Answer 21

That's the formula I used at first and I got \[\frac{ 5 }{ 2 }(1+ \frac{ 1 }{ 256 })\]

Answer 22

u need to use this formula only, for a geometric sequence
\(\huge S_n =a_1 \dfrac{1-r^n}{1-r} \)

Answer 23

your formula was right haha that's the right one ^

Answer 24

where a1 = 1st term = 1
n = number of terms = 5
and r= common ratio is what u need to find.

Answer 25

that would be appropriate for an arithmetic sequence, but your problem statement clearly says that this is a geom. seq.

ari97: would you please try hartnn's formula for the sum of the first n terms of a geom. seq.?

Answer 26

I'll do that one sec

Answer 27

before starting to find sum, find common ratio 'r'
know what that is ?

Answer 28

Note: early on I asked you: "Let's jump right in. The first term of this geometric sequence is what? a = ?
The common ratio is what? r = ?"

Answer 29

@hartnn: Many thanks for your help here.

Answer 30

I need a minute, I'm slow. Hold on

Answer 31

would r = 1/4?

Answer 32

Hint: if you do this problem on a calculator, use lots of parentheses to keep everything in the proper order and indicate proper order of operations.

Answer 33

Let me turn that around and ask you a question. If you take the 2nd term of your sequence, 1/4, and multiply that by r = 1/4, what would you get? Does that agree with the given sequence?

Answer 34

yess

Answer 35

Cool. Then your r= 1/4 is correct!

Answer 36

now let me plug in the equation, hold on

Answer 37

And your first term is a = ?

Answer 38

1

Answer 39

Great, and so the sum of the first 5 terms of this geometric sequence is ... ?

Answer 40

\[Sn = 1(\frac{ 1-5 }{ 1- \frac{ 1 }{ 4} })\] would this be correct?

Answer 41

Mind explaining where that 5 came from?

Answer 42

the number of terms right?

Answer 43

\[\huge S_n =a_1 \dfrac{1-r^n}{1-r}\]Does your expression agree with this formula?

Answer 44

Your denominator is fine! Your numberator needs attention.

Answer 45

I thought it dd, I'm confused :( hate fractions

Answer 46

1 - 1/4 is fine; that equals 3/4.

Answer 47

OH wait my numerator would be 1- 1/4^5 right?

Answer 48

a=1, so let's ignore a, writing 1 instead.

Answer 49

That's right: your numerator would be \[1-\frac{ 1 }{ 4^5 }\]

Answer 50

so the answer would be 341/256 I got it. THANK YOU SO MUCH OMG

Answer 51

and if you'd use your calculator to evaluate 4^5, you'll find that you have\[1-\frac{ 1 }{ 1024 }\] but don't take my word for this... check it.

Answer 52

Congrats. Very happy to work with you. Hope to meet you again soon on OpenStudy. :)

Answer 53

Thank you :)