Inductive Reasoning

Find the next number in the sequence $1,2,4,7,11$ by inductive reasoning.

Solution:

Observe: We see the sequence is increasing.

Pattern:

Sequence Pattern, Mouli Javia - StudySmarter Originals

Here the number increases by $1,2,3,4$ respectively.

Conjecture: The next number will be 16, because $11+5=16.$

Derive a conjecture for three consecutive numbers and test the conjecture.

Solution:

Consider groups of three consecutive numbers. Here these numbers are integers.

$1,2,3;5,6,7;10,11,12$

To make a conjecture, we first find a pattern.

$1+2+3;5+6+7;10+11+12$

Pattern: $1+2+3=6\Rightarrow 6=2\times 3$

$$\begin{gathered} 5+6+7=18\Rightarrow 18=6\times 3 \\ 10+11+12=33\Rightarrow 33=11\times 3 \end{gathered}$$

As we can see this pattern for the given type of numbers, let’s make a conjecture.

Conjecture: The sum of three consecutive numbers is equal to three times the middle number of the given sum.

Now we test this conjecture on another sequence to consider if the derived conclusion is in fact true for all consecutive numbers.

Test: We take three consecutive numbers $50,51,52.$

$50+51+52=153\Rightarrow 153=51\times 3$

The difference between two numbers is always less than its sum. Find the counterexample to prove this conjecture false.

Solution:

Let us consider two integer numbers say -2 and -3.

Sum: $(-2)+(-3)=-5$

Difference: $$\begin{gathered} (-2)-(-3)=-2+3=1 \\ \therefore 1\nless -5 \end{gathered}$$

Here the difference between two numbers –2 and –3 is greater than its sum. So, the given conjecture is false.

Make a conjecture about a given pattern and find the next one in the sequence.

Inductive reasoning sequence example, Mouli Javia - StudySmarter Originals

Solution:

Observation: From the given pattern, we can see that every quadrant of a circle turns black one by one.

Conjecture: All quadrants of a circle are being filled with color in a clockwise direction.

Next step: The next pattern in this sequence will be:

Next figure in sequence, Mouli Javia - StudySmarter Originals

Make and test conjecture for the sum of two even numbers.

Solution:

Consider the following group of small even numbers.

$2+8;10+12;14+20$

Step 1: Find the pattern between these groups.

$$\begin{gathered} 2+8=10 \\ 10+12=22 \\ 14+20=34 \end{gathered}$$

From the above, we can observe that the answer of all the sums is always an even number.

Step 2: Make a conjecture from step 2.

Conjecture: The sum of even numbers is an even number.

Step 3: Test the conjecture for a particular set.

Consider some even numbers, say, $68,102.$

The answer to the above sum is an even number. So the conjecture is true for this given set.

To prove this conjecture true for all even numbers, let’s take a general example for all even numbers.

Step 4: Test conjecture for all even numbers.

Consider two even numbers in the form: $x=2m,y=2n$, where $x,y$ are even numbers and $m,n$ are integers.

$x+y=2m+2n=2(m+n)$

Hence, it is an even number, as it is a multiple of 2 and $m+n$ is an integer.

So our conjecture is true for all even numbers.

Show a counterexample for the given case to prove its conjecture false.

Two numbers are always positive if the product of both those numbers is positive.

Solution:

Let us first identify the observation and hypothesis for this case.

Observation: The product of the two numbers is positive.

Hypothesis: Both numbers taken must be positive.

Here, we have to consider only one counterexample to show this hypothesis false.

Let us take into consideration the integer numbers. Consider –2 and –5.

$(-2)\times (-5)=10$

Here, the product of both the numbers is 10, which is positive. But the chosen numbers –2 and –5 are not positive. Hence, the conjecture is false.