Solve the problem.
-After being struck with a hammer, a gong vibrates 21 vibrations in the first second and in each second thereafter makes $$\frac { 3 } { 4 }$$ as many vibrations as in the previous second. Find how many vibrations the gong makes before it stops vibrating.
A)94 vibrations
B)28 vibrations
C)23 vibrations
D)84 vibrations

Question

Quizplus · Accepted Answer

The gong's vibrations form a geometric series with the first term $a = 21$ and common ratio $r = \frac{3}{4}$. The series continues until the vibrations stop, effectively an infinite series since the gong never truly stops vibrating but approaches zero. The sum $S$ of an infinite geometric series is given by $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio. Substituting $a = 21$ and $r = \frac{3}{4}$, we get $S = \frac{21}{1 - \frac{3}{4}} = \frac{21}{\frac{1}{4}} = 21 \times 4 = 84$.

Solve the Problem 34\frac { 3 } { 4 }43​ As Many Vibrations as in the Previous Second

Solve the Problem $\frac { 3 } { 4 }$ As Many Vibrations as in the Previous Second