What Comes Next?

In a recursive pattern, repetition of a rule or procedure can be used to extend the sequence or to find the values of any terms missing from the sequence.

Given a sequence of numbers, how can we discover the pattern and find the next number in the sequence? There are some common sequences that are straightforward to recognize, such as the sequence of positive integers

$$1, 2, 3, 4, 5, \ldots$$

or the sequence of odd prime numbers

$$3, 5, 7, 11, 13, 17, \ldots.$$

But what if we are given a more complicated sequence? What deductive approaches should we attempt to discover the sequence pattern? First, asking a few basic questions might be helpful. Is the sequence always increasing, or always decreasing, or neither? Is the sequence always positive, or always negative, or neither? Once we have answered these questions, here are a few methods to try to find the underlying sequence pattern:

1. Look at successive terms and take their differences (or ratios): is there a clear pattern in the differences (or ratios)?
2. Try adding another sequence (often constant or linear) to the current sequence.
3. Divide out by common factors.
4. Be aware of common patterns in the sequence itself (or in the differences or ratios of successive terms), such as integers, primes, factorial, arithmetic or geometric progressions, etc.

Draw the next term in this pattern.

ANSWER

In drawing the answer, the boxes could be added to the third term in two different locations: a horizontal row of five squares under the existing squares

or a diagonal line of five squares created by adding one box to the end of each existing row and starting a new row at the top of the image.

In either case, the resulting image is the same, and five squares were added to the pattern.

The same principles can be used to fill in the blanks if a term in a pattern is missing.

Draw the missing image from the following pattern:

ANSWER

Fill in the missing number $x$ in the following pattern: $1,4,9,x,25,36,\ldots$.

ANSWER

The pattern should read $1,4,9,16,25,36,\ldots$.

In analyzing this sequence, you may have noticed the values were perfect squares. Depending on how you solved the previous example, you may also have noticed that each value corresponds to the total number of small triangles in the pattern shown above. If you did not know what square numbers were, or did not notice the pattern, you could have counted the number of triangles in the fourth image if you noticed that pattern. Often, there are multiple ways of defining a recursive pattern and solving for missing terms. $_\square$

What comes next in the sequence $2,4,6,8, \dots?$

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Looking at the sequence, we recognize the even numbers, each of which is 2 more than the one before it. Thus the answer is $10$. $_\square$

What comes next in the sequence $1, 2, 5, 10, 17, 26, \dots?$

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Looking at the sequence carefully, we might decide to record the difference between terms:

$$\begin{aligned} 2-1 &= 1 \\ 5-2 &= 3 \\ 10-5 &= 5 \\ 17-10 &= 7 \\ 26-17 &= 9 . \end{aligned}$$

Thus we can see that the differences between adjacent terms in the sequence are the odd numbers and that the next difference must be 11. Therefore the next term in the sequence is $26 + 11 = 37$. $_\square$

What comes next in the sequence $50, 49, 47, 44, 40, 35, \dots?$

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First, notice that this sequence is decreasing. Taking the difference between successive terms, we obtain

$$\begin{aligned} 50-49 &= 1 \\ 49-47 &= 2 \\ 47-44 &= 3 \\ 44-40 &= 4 \\ 40-35 &= 5. \end{aligned}$$

From these differences, we see that terms in the sequence differ by increasing integers, and the next difference is 6. Therefore, the next term in the sequence is $35-6 = 29$. $_\square$

What comes next in the sequence $1, -3, 9, -27, 81, \dots?$

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This sequence is alternating between positive and negative terms, and the first terms are multiples of $3$. This suggests that we might try taking the ratio of successive terms, which gives

$$\begin{aligned} \frac{-3}{1} &= -3 \\\\ \frac{9}{-3} &= -3 \\\\ \frac{-27}{9} &= -3 \\\\ \frac{81}{-27} &= -3. \end{aligned}$$

From these ratios, we see that the ratio between successive terms are all $-3$. Therefore, the next term in the sequence is $(81) \cdot (-3) = -243$. $_\square$

What comes next in the sequence $1, 3, 7, 15, 31, 63, \dots?$

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This sequence is increasing and the gaps between successive terms is increasing. If we take the difference between successive terms, we obtain

$$\begin{aligned} 3-1 &= 2 \\ 7-3 &= 4 \\ 15-7 &= 8 \\ 31-15 &= 16 \\ 63 - 31 &= 32. \end{aligned}$$

We recognize these differences as powers of $2.$ Therefore, the next term in the sequence is $63 + 2^6 = 63 + 64 = 127$.

Alternatively, we can notice that the terms in the sequence are all odd, and adding 1 to each term in the sequence gives $2, 4, 8, 16, 32, 64 \dots$. Therefore, the next term in the sequence is $128-1=127$ and, in general, term $n$ in the sequence is given by $2^n -1$. $_\square$

What comes next in the sequence $1, 2, 4, 8, 16, 32, 64, \dots?$

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If you carefully observe, you will see that these numbers are the powers of $2$. Therefore, the numbers are $2^0, 2^1, 2^2, 2^3, 2^4, 2^5, 2^6, \ldots.$ Hence, the next number in this sequence should be $2^7,$ which equals $128. \ _\square$