Math  /  Discrete

QuestionUse inductive reasoning to predict the next three numbers in the pattern (or sequence). 4,8,13,19,26,34,4,8,13,19,26,34, \ldots
The next three numbers in the pattern, in order, are \square \square , and \square (Simplify your answers.)

Studdy Solution

STEP 1

What is this asking? We need to figure out the next three numbers in a sequence, and we're given a hint to use "inductive reasoning," which just means finding the pattern! Watch out! Be careful not to jump to conclusions too quickly!
Sometimes patterns can be tricky, so make sure you test your theory on a few numbers before you're absolutely sure.

STEP 2

1. Find the differences between consecutive numbers.
2. Identify the pattern in the differences.
3. Predict the next differences.
4. Calculate the next three numbers.

STEP 3

Let's look at the differences between each number and the one before it.
This helps us see how the sequence is changing.
The difference between 88 and 44 is 84=48 - 4 = 4.
The difference between 1313 and 88 is 138=513 - 8 = 5.
Keep going!

STEP 4

The difference between 1919 and 1313 is 1913=619 - 13 = 6.
The difference between 2626 and 1919 is 2619=726 - 19 = 7.
The difference between 3434 and 2626 is 3426=834 - 26 = 8.

STEP 5

Look at the differences we found: 4,5,6,7,84, 5, 6, 7, 8.
Do you see a pattern?
It looks like the differences are increasing by **one** each time!

STEP 6

If the pattern continues, the next difference should be 8+1=98 + 1 = 9, then 9+1=109 + 1 = 10, and then 10+1=1110 + 1 = 11.

STEP 7

To find the next number in the sequence, we add the next predicted difference to the last number in the original sequence.
So, 34+9=4334 + 9 = 43.
That's our first predicted number!

STEP 8

The next number is found by adding 1010 to the previous result: 43+10=5343 + 10 = 53.
Awesome!

STEP 9

And the final predicted number is 53+11=6453 + 11 = 64.
We did it!

STEP 10

The next three numbers in the sequence are 4343, 5353, and 6464.

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