Use inductive reasoning to predict the next equation.
-$\begin{array} { r }
\frac { 1 } { 3 } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 3 } \right) \
\frac { 1 } { 3 } + \frac { 1 } { 9 } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 9 } \right) \
\frac { 1 } { 3 } + \frac { 1 } { 9 } + \frac { 1 } { 27 } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 27 } \right) \
\frac { 1 } { 3 } + \frac { 1 } { 9 } + \frac { 1 } { 27 } + \frac { 1 } { 81 } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 81 } \right)
\end{array}$

A) $\frac { 1 } { 3 } + \frac { 1 } { 9 } + \frac { 1 } { 27 } + \frac { 1 } { 81 } + \frac { 1 } { 243 } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 243 } \right)$
B) $\frac { 1 } { 3 } + \frac { 1 } { 9 } + \frac { 1 } { 27 } + \frac { 1 } { 81 } + \frac { 1 } { 729 } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 729 } \right)$
C) $\frac { 1 } { 3 } + \frac { 1 } { 9 } + \frac { 1 } { 27 } + \frac { 1 } { 81 } + \frac { 1 } { 162 } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 162 } \right)$
D) $\frac { 1 } { 3 } + \frac { 1 } { 9 } + \frac { 1 } { 27 } + \frac { 1 } { 81 } + \frac { 1 } { 243 } = \frac { 1 } { 3 } \left( 1 - \frac { 1 } { 243 } \right)$

Question

Use inductive reasoning to predict the next equation.
-$\begin{array} { r } 
\frac { 1 } { 3 } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 3 } \right) \
\frac { 1 } { 3 } + \frac { 1 } { 9 } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 9 } \right) \
\frac { 1 } { 3 } + \frac { 1 } { 9 } + \frac { 1 } { 27 } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 27 } \right) \
\frac { 1 } { 3 } + \frac { 1 } { 9 } + \frac { 1 } { 27 } + \frac { 1 } { 81 } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 81 } \right)
\end{array}$

A) $\frac { 1 } { 3 } + \frac { 1 } { 9 } + \frac { 1 } { 27 } + \frac { 1 } { 81 } + \frac { 1 } { 243 } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 243 } \right)$
B) $\frac { 1 } { 3 } + \frac { 1 } { 9 } + \frac { 1 } { 27 } + \frac { 1 } { 81 } + \frac { 1 } { 729 } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 729 } \right)$
C) $\frac { 1 } { 3 } + \frac { 1 } { 9 } + \frac { 1 } { 27 } + \frac { 1 } { 81 } + \frac { 1 } { 162 } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 162 } \right)$
D) $\frac { 1 } { 3 } + \frac { 1 } { 9 } + \frac { 1 } { 27 } + \frac { 1 } { 81 } + \frac { 1 } { 243 } = \frac { 1 } { 3 } \left( 1 - \frac { 1 } { 243 } \right)$

Quizplus · Accepted Answer

Each term on the left side of the equations is a fraction with a denominator that is a power of 3, increasing by one power with each new term. The right side of the equation is always $\frac{1}{2}(1 - \text{the last term's denominator as a fraction})$. The next term after $\frac{1}{81}$ is $\frac{1}{243}$, following the pattern of powers of 3 ($3^5 = 243$), and the equation structure remains consistent with choice A.

Use Inductive Reasoning to Predict the Next Equation A) B)