Answer:
Series converges
Sum = 192
Step-by-step explanation:
General form of geometric series: [tex]a_n=ar^{n-1}[/tex]
where:
- a is the initial term
- r is the common ratio
Given series: 48, 36, 27, 81/4, ...
[tex]\implies a = 48[/tex]
[tex]\implies r=\dfrac{a_2}{a_1}=\dfrac{36}{48}=\dfrac34[/tex]
Geometric series converges when |r| < 1
Geometric series diverges when |r| ≥ 1
[tex]\textsf{As }r=\dfrac34\ \implies|\dfrac34| < 1\:\implies\:\textsf{series converges}[/tex]
Sum of an infinite geometric series:
[tex]S_\infty=\dfrac{a}{1-r}\quad\textsf{for}\:|r| < 1[/tex]
Substituting values of a and r:
[tex]\implies S_\infty=\dfrac{48}{1-\frac34}=192[/tex]
