Question

Find the sum of each finite geometric series.

1+2+4+...+2048

Answer 1

The common ratio r is 2 in this case. When the common ratio is positive and greater than 1 the formula for the sum of n terms is
\[S _{n}=\frac{a(r ^{n}-1)}{r-1}\]
For this question we need to find the value of n for the last term 2048.
The nth term is found from the formula
\[n ^{th} term=ar ^{n-1}\]
Plugging in the known values of a and r gives
\[2048=1\times2^{n-1}\ .........(1)\]
2 to the power of 11 = 2048. Therefore n = 11 - 1 = 10
Now you just need to plug the values of a, r and n into the formula for the sum of n terms.

Answer 2

the last part is the part I don't understand, can you explain it simply what I have to do "to plug in"

Answer 3

Sorry, my bad. The final part of my post should be:
"2 to the power of 11 = 2048. Therefore n = 11 + 1 = 12. Now you just need to plug the values of a, r and n into the formula for the sum of n terms."
The values are as follows:
a = 1
r = 2
n = 12
The formula is
\[S _{n}=\frac{a(r ^{n}-1)}{r-1}\]

Answer 4

I thought it was n-1 so wouldn't it be 11 - 1 instead of 11 + 1

Answer 5

\[2^{11}=2048\]
\[2^{n-1}=2048\]
Therefore n - 1 = 11 ........(2)
Adding 1 to both sides of equation (2) gives
n = 11 + 1 = 12

Answer 6

oh okay!

Answer 7

can you work me through the formula?

Answer 8

The values are as follows: a = 1 r = 2 n = 12 The formula is
\[S _{n}=\frac{a(r ^{n}-1)}{r-1}\]
So after plugging in the values we get
\[S _{12}=\frac{1(2^{12}-1)}{2-1}=you\ can\ calculate\]

Answer 9

that's actually the part I don't get, don't understand how to calculate it :(

Answer 10

2^18 = 262144?

Answer 11

Where did you get '2 to the power of 18' from?

Answer 12

oh gosh, I misread it.

Answer 13

oh gosh, I misread it.

Answer 14

the 12 looked like an 18 to me, I'm sorry, haha

Answer 15

so 4096

Answer 16

When you simplify the fraction you get
\[S _{12}=\frac{1(2^{12}-1)}{2-1}=2^{12}-1\]

Answer 17

4095 is it then?

Answer 18

4095 is correct!

Answer 19

thank you so much! I actually think I'm starting to get it, I just need nudges in the right direction! :)

Answer 20

You're welcome :) Glad to hear you are understanding.