Use inductive reasoning to find the solution to the problem; then check it by performing the calculation.

$\begin{aligned} &0 \cdot 9+8=8 \\ &9 \cdot 9+7=88 \\ &98 \cdot 9+6=888 \\ &987 \cdot 9+5=8,888 \\ &\vdots \\ &9,876,543 \cdot 9+1=? \end{aligned}$

Solution

VerifiedHere, we have given the following equations.

$0\cdot 9 +8=8$

$09\cdot 9 +7=88$

$098\cdot 9 +6=888$

$0987\cdot 9 +5=8888$

We need to observe a pattern between these equations and guess the right side of the equation $(09876543\cdot 9 + 1)$. To find the pattern we can write the given equations as follow.

$0(\text{one digit})\cdot 9 +8=8(\text{one time})$

$09(\text{two digits})\cdot 9 +7=88(\text{two times})$

$098(\text{three digits})\cdot 9 +6=888(\text{three times})$

$0987(\text{four digits})\cdot 9 +5=8888(\text{four times})$

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