Question

Find the sum of each of the geometric series given below. For the value of the sum, find an expression that gives the exact value, rather than giving an approximation.

1) -6 + 2 - (2/3) + (2/9) - (2/27) + (2/81) - ...

2)(Sigma from n=5 to n=12) (1/3)^n

Answer 1

bah this is to hard on my tablet, brb

Answer 2

ok the first term is -6 and the common ratio is -1/3
so we have

-6/(1-(-1/3)) = -9/2

Answer 3

\(\sum_{0}^\infty ar^n=\frac{a}{1-r}\) as long as \(|r|<1\)

Answer 4

With r being the ratio.

Answer 5

your first one is \(\sum_0^\infty(-1)^{n+1}6(\frac{1}{3})^n\)

Answer 6

correct

Answer 7

what is the ratio of the second one?

Answer 8

Hold on. Why not make the first one similar, but with this as the equation instead: -6(-1/3)^(n-1)

Answer 9

Because with the first number in the series, that would make it positive two, instead of negative 6, correct?

Answer 10

you can do that.
to find the sum of infinite geometric series, we need 2 things:
1) first term
2) common ratio
no matter how you find it.

so whats first term and ratio for 2nd part ?

Answer 11

First term would be 1/3, with a ratio of 1/3?

Answer 12

For the second problem, correct?

Answer 13

oops, i din't note that 2nd problem is not of infinite sequence

its finite, n= 5 to 12

Answer 14

Yeah, I was a bit confused there! Lol, I ended up getting the first one, but this second one is giving me problems.

Answer 15

the formula you use is :
\(\large S_n=a_1\frac{(1-r^n)}{(1-r)}\)

a1 = 1st term ,
n= number of terms
r = ratio = 1/2

to get first term, put n=5 in (1/3)^n
because n starts from 5

Answer 16

to get number of terms,
well, from 5 to 12, there are
12-5+1 terms
so your n in the formula will be n =12-5+1

Answer 17

Six terms. Gotcha.

Answer 18

Ended up not getting it quite right. I'll try to work on it tomorrow, letting this thread sit overnight.

Answer 19

12-5 is 7 and 7+1 is 8

5,6,7,8,9,10,11,12

n=8
a1 = (1/3)^5
r =1/3
:)

Answer 20

Bah! I feel dumb now. Did my addition and subtraction wrong. D: Thank you!