Answer:
Further explanation
Geometry sequences are series of numbers that have a constant ratio
or can be interpreted:
Each number is obtained by multiplying the previous number by a constant
The sequence can be:
a, ar, ar², ar³,ar⁴ ... etc.
Can be formulated
\large{\boxed{\bold{x_n=a.r^{(n-1)}}}
where:
a is the first term, and
r is the common ratio
For Infinite Geometric Series
\boxed{\bold{S\sim=\dfrac{a}{1-r}}}\Rightarrow\:\bold{|r| < 1}\Rightarrow convergent
S∼=
1−r
a
⇒∣r∣<1⇒convergent
so r not including −1 and 1 (-1 < r < 1) and If r> 1, it is said that the series is divergent which cannot be determined
1. a = 64, r = 1/4
\begin{gathered}\rm S\sim=\dfrac{64}{1-\dfrac{1}{4} }\\\\S\sim=\dfrac{64}{\dfrac{3}{4} }\\\\S\sim=\boxed{\bold{\dfrac{256}{3}}}\end{gathered}
S∼=
1−
4
1
64
S∼=
4
3
64
S∼=
3
256
2. a = 1/3, r = 1/3
\begin{gathered}\rm S\sim=\dfrac{\dfrac{1}{3} }{1-\dfrac{1}{3} }\\\\S\sim=\dfrac{\dfrac{1}{3} }{\dfrac{2}{3} }\\\\S\sim=\boxed{\bold{\dfrac{1}{2}}}\end{gathered}
S∼=
1−
3
1
3
1
S∼=
3
2
3
1
S∼=
2
1
3. a = -4, r = 1/4
\begin{gathered}\rm S\sim=\dfrac{-4}{1-\dfrac{1}{4} }\\\\S\sim=\dfrac{-4}{\dfrac{3}{4} }\\\\S\sim=\boxed{\bold{\dfrac{-16}{3}}}\end{gathered}
S∼=
1−
4
1
−4
S∼=
4
3
−4
S∼=
3
−16
4. a = 24, r = 1/6
\begin{gathered}\rm S\sim=\dfrac{24}{1-\dfrac{1}{6} }\\\\S\sim=\dfrac{24}{\dfrac{5}{6} }\\\\S\sim=\boxed{\bold{\dfrac{144}{5}}}\end{gathered}
S∼=
1−
6
1
24
S∼=
6
5
24
S∼=
5
144
5. a = 1, r = √2
Because r> 1, and the value of the number is getting bigger so the sum can not be determined (divergent series)
