Question

For each exercise, write the equation you think should come next. Check your prediction with a calculator.

$\begin{array}{l}{9}^2=81\\ {99}^2=9801\\ {999}^2=998001\end{array}$

Short answer

The next equation must be ${9999}^2=99980001$.

Step-by-step solution

Step 1. Define deductive and inductive reasoning.

Deductive reasoning is conclusions based on accepted statements like a theorem, postulate, facts etc.

Inductive reasoning is conclusions based on several past observations.

Step 2. Look for a pattern.

While looking for a pattern in the sequence

$\begin{array}{l}{9}^2=81\\ {99}^2=9801\\ {999}^2=998001\end{array}$

reason inductively that is., we can conclude based on past observations

As,

$\begin{array}{r}9\to 99\to 999\\ 81\to 9801\to 998001\end{array}$

raised to the power square is fixed.

Step 3. Predict the next equation.

So, as per the pattern of the sequence, next terms in the equation will be:

$\begin{array}{r}9\to 99\to 999\to \underset{¯}{9999}\\ 81\to 9801\to 998001\to \underset{¯}{99980001}\end{array}$

Therefore, the next equation must be ${9999}^2=99980001$.