Number and Figure Series: Patterns, Methods and Practice

1. What this chapter teaches#

A series is an ordered list of terms. A question asks you to discover a rule that explains the supplied terms and then apply it accurately. There are two different ways to classify a series:

The same multiplication pattern can appear as a missing-term question or a wrong-term question. First identify the format, then discover the rule. For a missing term inside a sequence, the rule must fit terms on both sides of the gap. For a wrong term, replacing one term must repair the whole sequence.

This chapter covers arithmetic and multilevel differences; multiplication and division; squares, cubes and offsets; primes and Fibonacci recurrences; alternating and mixed operations; coded double series; and nonverbal figure series. It also adds factorials, triangular numbers and digit patterns because these often resemble the listed patterns at first glance.

Important reasoning limit: a short finite list can fit more than one mathematical rule. In an exam, choose the simplest natural rule that explains every given term and the available answer options. Do not claim that a pattern is uniquely proved from a few terms alone.

2. A reliable six-step solving method#

1. Read the task. Is the blank at the end or in the middle? Does the question ask for a wrong term, a pair of missing terms, or a figure?
2. Look at size and direction. Slow steady change suggests addition; rapid growth suggests multiplication or powers; rapid decline suggests division. These are clues, not proofs.
3. Write first differences $d_i=a_{i+1}-a_i$. If they are not constant, check the differences of those differences. For rapidly growing nonzero terms, also inspect ratios $a_{i+1}/a_i$.
4. Test familiar families. Compare with $n^2$, $n^3$, primes, sums of the previous two terms, factorials and triangular numbers. Check a simple offset such as $n^2+1$.
5. Split positions when needed. Read terms in positions 1, 3, 5, ... separately from positions 2, 4, 6, .... A single row can hide two simple series.
6. Verify the complete row. Calculate the rule across all known transitions, then apply it to the requested place. If it fails even one supplied term, look again.

For a timed question, do quick differences and ratios first, then test position splitting and familiar number families. When several rules fit, use the intended simple pattern and any answer choices to decide. Never extend a rule from only its last two terms while ignoring the beginning.

3. Arithmetic and difference series#

In an arithmetic progression, the first difference is fixed: $a_{n+1}=a_n+d$. The common difference $d$ can be positive, zero or negative.

Not every difference series has one fixed difference. In $1, 4, 9, 16, 25$, the gaps are $3, 5, 7, 9$; the next gap is $11$, giving $36$. This can also be recognised directly as consecutive squares. State the rule that explains the whole list most clearly.

For a middle blank, use both neighbours. In $8, 13, ?, 23$, a $+5$ rule gives $18$, and $18+5=23$ verifies the right side.

4. Second and higher differences#

When first differences vary regularly, take another row of differences. For $2, 5, 10, 17, 26, ?$:

The next term is $26+11=37$. The same terms satisfy $a_n=n^2+1$ for $n=1,2,3,\ldots$. A fixed second difference is typical of a quadratic pattern. A fixed third difference is typical of a cubic pattern: the cubes $1,8,27,64,125$ have first differences $7,19,37,61$, second differences $12,18,24$, and third differences $6,6$.

The procedure is: make first differences → make second differences → extend the simplest regular difference row → work back upward. Do not keep taking many difference levels just to force an arbitrary answer. With too few given terms, high-order differences can manufacture a rule that is less useful than an obvious square or cube pattern.

5. Geometric and multiplication series#

If each term is multiplied by the same nonzero number $r$, the series is geometric: $a_{n+1}=r a_n$. For example, $3,6,12,24,?$ uses $\times 2$ and gives $48$.

Multipliers themselves can change regularly. In $3,6,18,72,360,?$ the multipliers are $2,3,4,5$; continue with $\times 6$ to get 2160. Write the multipliers over the arrows before guessing from the size of the terms.

Negative multipliers can make signs alternate: $2,-4,8,-16,?$ uses $\times(-2)$, so the next term is $32$. A zero term breaks ratio checking because division by zero is undefined; inspect another feature instead.

6. Division and decreasing series#

For $243,81,27,9,?$ each term is divided by $3$, so the answer is 3. In $720,360,120,30,6,?$ the divisors are $2,3,4,5$; the next divisor is $6$, giving 1.

Division can produce fractions. For example, $8,4,2,1,?$ continued by $\div 2$ gives $\tfrac12$. Do not assume the answer must be an integer unless the question or options require it. A decreasing row can also be repeated subtraction, alternating operations or reversed powers, so test the ratios rather than relying on appearance.

7. Squares, cubes and shifted powers#

Memorise small squares and cubes well enough to recognise them without repeatedly calculating differences.

Check the starting index. A list beginning $4,9,16$ could be squares of $2,3,4$, rather than the $n=1$ row. Sometimes terms are $2^n$ or $3^n$ rather than $n^2$ or $n^3$: $2,4,8,16$ doubles, while $1,4,9,16$ consists of squares. Test all terms before choosing.

8. Prime-number series#

A prime is an integer greater than 1 with exactly two positive factors: 1 and itself. The first primes are $2,3,5,7,11,13,17,19,23,29$. The number 1 is not prime, and 2 is the only even prime.

For $2,3,5,7,11,13,?$ the answer is 17, not 15. The gaps between consecutive primes are not fixed, so a simple difference extrapolation is unreliable here. Some questions add or subtract a fixed number from each prime: $3,4,6,8,12,14,?$ is each of the first primes plus 1, giving 18. Check the prime basis term by term.

9. Fibonacci and other two-term recurrences#

In a Fibonacci-style recurrence, each term is the sum of the previous two: $a_n=a_{n-1}+a_{n-2}$. The familiar row $1,1,2,3,5,8,13,?$ continues to 21. A row may instead start $0,1$ or with entirely different seeds, such as $2,4,6,10,16,26,?$ → 42.

Always check at least three consecutive sum relationships. Seeing $2+3=5$ once is insufficient to label a series Fibonacci. Some harder rows use $a_n=a_{n-1}+a_{n-2}+c$ or alternate the recurrence with another operation; write each transition explicitly.

10. Alternate or twin series#

A twin series hides two independent sequences in odd and even positions. Split the row before trying a complicated single formula.

The odd subsequence adds 2; the even subsequence adds 10. Therefore the eighth term is 40. In the harder row $1,2,4,6,9,12,16,?$ the odd positions are $1,4,9,16$ (squares), while even positions are $2,6,12,20$ ($n(n+1)$). The missing value is 20.

An alternating pattern can also switch operations: $2,4,7,14,17,34$ alternates $\times 2$ and $+3$. Check whether the next operation is $+3$ or $\times 2$ by tracking the exact position.

11. Mixed and multi-operation series#

Mixed patterns combine operations. The rule may repeat, or the added/subtracted value may change by one each step.

In $2,5,12,27,58,?$:

The answer is 121. A different list may use $\times 3-1$, then $\times 3-2$, and so on. Identify both the multiplier and the changing adjustment; saying only “multiply by 2” does not explain the displayed terms.

12. Other useful patterns: triangular numbers, factorials and digits#

These extra families strengthen pattern recognition in difficult questions:

Digit reversal only makes sense if the whole row supports it; do not invent a digit trick for an ordinary multiplication row. Factorials grow very quickly and can resemble a variable-multiplier series; in fact the multiplier test reveals the factorial structure.

13. Missing-term questions#

For an end blank, derive the next operation. For a middle blank, verify the operation into and out of the missing place. Example: $5,10,?,40,80$ under doubling gives $20$ because $10\times2=20$ and $20\times2=40$.

In an alternating series, determine whether the blank occupies an odd or even position before extrapolating. If options are supplied, substitute each promising option into the entire row, not just its nearest neighbours. A visually plausible number may break an earlier transition.

14. Wrong-term questions#

One displayed number is inconsistent with the intended pattern. Replace exactly one term and require all the other terms to fit.

Example: $2,4,8,16,31,64$. Repeated doubling requires $2,4,8,16,32,64$, so 31 is wrong; 32 is the correction. Do not report 64 as wrong merely because $31\times2\ne64$; correcting 31 repairs both adjacent transitions.

Example: $1,4,9,15,25,36$. The square at position 4 should be $16$, so 15 is wrong. A bad term often disturbs two nearby first differences. Examine the wider sequence before deciding which of those positions contains the error.

15. New-pattern or coded double series#

A double-series question gives a complete or nearly complete Series I and asks you to transfer its rule to Series II. The rule may be a fixed operation, a changing operation, or an index-based construction. It is learned from Series I; there is no universal “coded-series formula”.

Worked transfer: Series I: $2,5,11,23,47$. The transitions are $\times2+1$ every time. Apply the same operation to Series II: $3,7,15,?,63$. Since $15\times2+1=31$ and $31\times2+1=63$, the blank is 31.

Changing-rule transfer: Series I: $1,3,6,10,15$ has increments $+2,+3,+4,+5$. Series II: $4,6,9,13,18,?$ follows the same ordered increments; after $+5$ comes $+6$, so the answer is 24.

Keep the operation sequence aligned by transition number. Do not copy Series I's actual term values into Series II. Verify that the transferred rule fits every provided term of both rows.

16. Nonverbal or figure series#

Figure-series questions replace numbers with shapes. Inspect one feature at a time: rotation, direction, position, count, shading, size, reflection and alternation. A symbol may vary in more than one feature, so verify the full progression.

Each triangle turns a quarter-turn clockwise. Therefore position 6 is ▶. Here the triangle glyphs stand for the same figure at different orientations.

For $●, ●●, ●●●, ?$, one dot is added each time, giving ●●●●. For $●, ○, ●, ○, ?$, filling alternates, giving ●. If an exam shows overlapping shapes, count their components carefully; a mirror image reverses left and right, while a rotation changes orientation around a centre.

When figures are printed at different apparent sizes, check whether size is actually part of the rule. Do not mistake font rendering differences for a logical change.

17. Compare candidates and avoid common traps#

Verification habit: write the proposed rule next to each transition. A rule that cannot reproduce an earlier term is not an adequate answer, even if it predicts one of the options. If two equally simple rules still fit all terms, the question is ambiguous without further constraints; use options or stated conventions rather than pretending certainty.

18. Ten fully worked examples#

Example 1 — descending arithmetic row: $52,45,38,31,?$. Every difference is $-7$, so the next term is 24.

Example 2 — second difference and squares: $4,9,16,25,36,?$. The differences are $5,7,9,11$; the next difference is $13$, giving 49. These terms are also consecutive squares beginning at $2^2$.

Example 3 — third difference and shifted cubes: $2,9,28,65,126,?$. These are $n^3+1$ for $n=1$ to $5$; the sixth term is $6^3+1=\mathbf{217}$. First differences $7,19,37,61$, second differences $12,18,24$, and third differences $6,6$ confirm the cubic pattern.

Example 4 — increasing multiplier: $5,10,30,120,600,?$. The multipliers are $2,3,4,5$; continue with $\times6$ to get 3600.

Example 5 — increasing divisor: $5040,2520,840,210,42,7,?$. Divide by $2,3,4,5,6$; next divide by $7$, giving 1.

Example 6 — one less than a square: $0,3,8,15,24,?$. The terms are $1^2-1,2^2-1,\ldots$, so the sixth is $6^2-1=\mathbf{35}$.

Example 7 — previous-two-term sum: $1,3,4,7,11,18,?$. Check $7+11=18$; the next term is $11+18=\mathbf{29}$.

Example 8 — alternating rows: $2,100,4,90,6,80,8,?$. The even-position row is $100,90,80,70$, so the missing value is 70.

Example 9 — fixed mixed operation: $1,2,5,14,41,?$. Each step is $\times3-1$; therefore $41\times3-1=\mathbf{122}$.

Example 10 — wrong term: $5,10,20,40,81,160$. Repeated doubling requires the fifth term to be $80$. Thus 81 is wrong; 80 is the correction, and $80\times2=160$ verifies it.

19. Practice: 20 questions, including harder patterns#

Find the missing term unless a question explicitly asks for a wrong term. Work each question first, then use the explanations below. For double series, find the requested blank in each series.

Practice 01#

$4,9,14,19,?$

Practice 02#

$2,5,10,17,26,?$

Practice 03#

$3,6,18,72,360,?$

Practice 04#

$720,360,120,30,6,?$

Practice 05#

$3,6,11,18,27,?$

Practice 06#

$1,8,27,64,?$

Practice 07#

$2,3,5,7,11,13,?$

Practice 08#

$2,4,6,10,16,26,?$

Practice 09#

$2,10,4,20,6,30,8,?$

Practice 10#

$2,5,12,27,58,?$

Practice 11#

$96,48,24,12,?$

Practice 12#

Find the wrong term and give its correction: $2,4,8,16,31,64$.

Practice 13#

Find the wrong term and give its correction: $1,4,9,15,25,36$.

Practice 14#

Find the wrong term and give its correction: $2,3,5,8,11,13$.

Practice 15#

Transfer one rule between the rows and find both blanks. Series I: $2,5,11,23,?$; Series II: $3,7,15,?,63$.

Practice 16#

Transfer the sequence of operations. Series I: $1,3,6,10,15$; Series II: $4,6,9,13,18,?$.

Practice 17#

Find the next figure: ▲, ▶, ▼, ◀, ▲, ?

Practice 18#

Find the next figure: ●, ●●, ●●●, ?

Practice 19#

$1,2,6,24,120,?$

Practice 20#

$1,2,4,6,9,12,16,?$

20. Answers with the complete reasoning#

Answer 01#

24. Every first difference is $+5$, so $19+5=24$.

Answer 02#

37. The gaps are $3,5,7,9$; next is $11$, hence $26+11=37$. Equivalently these are $n^2+1$.

Answer 03#

2160. The multipliers are $2,3,4,5$; next is $6$, so $360\times6=2160$.

Answer 04#

1. Divide successively by $2,3,4,5,6$: $6\div6=1$.

Answer 05#

38. The terms are $n^2+2$ for $n=1$ to $5$; the next is $6^2+2=38$. The gaps $3,5,7,9,11$ confirm it.

Answer 06#

125. These are $1^3,2^3,3^3,4^3$, followed by $5^3=125$.

Answer 07#

17. List consecutive primes after 13; the next is 17.

Answer 08#

42. Each term after the first two is their preceding pair's sum: $16+26=42$.

Answer 09#

40. Even-position terms are $10,20,30,40$; odd-position terms are $2,4,6,8$.

Answer 10#

121. Apply $\times2+1$, then $\times2+2$, $\times2+3$, $\times2+4$; next $58\times2+5=121$.

Answer 11#

6. Every term is halved: $12\div2=6$.

Answer 12#

31 is wrong; replace it with 32. Then every term doubles, including $32\times2=64$.

Answer 13#

15 is wrong; replace it with 16. The intended row is $1^2,2^2,3^2,4^2,5^2,6^2$.

Answer 14#

8 is wrong; replace it with 7. The corrected row is $2,3,5,7,11,13$, the first six primes.

Answer 15#

Series I: 47; Series II: 31. Both use $\times2+1$: $23\times2+1=47$ and $15\times2+1=31$. The latter also verifies $31\times2+1=63$.

Answer 16#

24. Series I increments are $+2,+3,+4,+5$. Series II has the same increments and next needs $+6$, so $18+6=24$.

Answer 17#

▶. Each triangle rotates clockwise by a quarter-turn: up, right, down, left, up, right.

Answer 18#

●●●●. The number of filled dots rises by one each time: one, two, three, four.

Answer 19#

720. Multiply by $2,3,4,5,6$ successively; $120\times6=720$. This is the factorial sequence.

Answer 20#

20. Odd-position terms are $1,4,9,16$ (squares). Even-position terms are $2,6,12,20$, following $n(n+1)$ for $n=1,2,3,4$.

21. One-page revision guide#

Fast recognition order: first differences → second differences → ratios → squares/cubes/primes → previous-two-term sum → odd/even split → mixed operations → extra families. For a decreasing series, test division as well as subtraction. For a figure series, track rotation, position, count and shading separately.

Question-format checks: a missing middle term must fit both neighbours; a wrong term must be the single change that repairs the row; a double series transfers the operation sequence; a figure series must respect the visual feature consistently. After finding an answer, recalculate at least the two transitions around it and scan the earlier terms for the same rule.

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