2.1 Linear, Polynomial, Alternating & Recursive Number Series (Sayı Dizileri)

Linear, Polynomial, Alternating & Recursive Number Series (Sayı Dizileri)

Practice Strategy: Number-series exercises build numerical pattern recognition. Check a proposed rule against every supplied transition, then compare the options. Record your own solving times instead of assuming these items occupy a fixed opening position or take a fixed number of seconds.

A Sayı Dizisi presents a finite sequence of numbers—typically 5 to 9 terms—governed by a deterministic generation law, asking you to determine the missing term ($?$) or a pair of missing terms. Read the instruction and any stated rule restrictions; do not assume that every sequence is presented without verbal information.

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The Five Core Archetypes of TR-YÖS Number Series

The following five families are a practical checklist, not an exhaustive classification. A finite list can fit multiple rules; use the stated constraints and the simplest rule consistent with all supplied terms.

1. First- and Second-Order Arithmetic Difference Series (Fark Dizileri)

When terms grow or shrink gradually at a steady or smoothly accelerating pace, write the first differences ($\Delta_1 = a_{n+1} - a_n$) directly beneath the sequence.

• First-Order Arithmetic Series ($\Delta_1 = d$): The difference between consecutive terms is constant. The general term is linear:

$$a_n = a_1 + (n - 1)d$$

• Second-Order Quadratic Series ($\Delta_2 = c$): The first differences themselves form an arithmetic progression ($\Delta_1$ increases or decreases by a constant $c$). The sequence is governed by a quadratic polynomial:

$$a_n = An^2 + Bn + C \quad \text{where } \Delta_2 = 2A$$

Two classic special cases appear repeatedly:

• Triangular Numbers (Üçgensel Sayılar): $T_n = \frac{n(n+1)}{2} \implies 1, 3, 6, 10, 15, 21, 28, 36, \dots$ ($\Delta_1 = +2, +3, +4, +5, \dots$)
• Pronic Numbers (Ardışık Çarpım Sayıları): $P_n = n(n+1) \implies 2, 6, 12, 20, 30, 42, 56, \dots$ ($\Delta_1 = +4, +6, +8, +10, \dots$)

Worked Example 1: Second-Order Difference Series

Find the next term in the sequence: $4, 7, 12, 19, 28, 39, ?$

1. Compute first differences ($\Delta_1$):

$$7 - 4 = +3, \quad 12 - 7 = +5, \quad 19 - 12 = +7, \quad 28 - 19 = +9, \quad 39 - 28 = +11$$

1. Compute second differences ($\Delta_2$): Each consecutive difference increases by $+2$ (the odd integers $+3, +5, +7, +9, +11$).
2. Extrapolate: The next first difference is $11 + 2 = +13$. Therefore:

$$? = 39 + 13 = 52$$

(Algebraic check: $a_n = n^2 + 3$ for $n = 1, 2, \dots, 7$, giving $7^2 + 3 = 52$.)

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2. Geometric & Mixed Multiplicative-Additive Series (Kat ve Eklemeli Diziler)

When terms grow rapidly—doubling or tripling at each step—first differences grow exponentially. Instead of pure subtraction, test a multiplicative-additive recurrence of the form:

$$a_{n+1} = r \cdot a_n \pm k$$

or an alternating two-operator cycle (such as $\times r$ followed by $+ k$, repeated).

• Constant Multiplier with Fixed Offset: In $3, 7, 15, 31, 63, ?$, each term is $a_{n+1} = 2a_n + 1$, so the next term is $2(63) + 1 = 127$. Notice that the first differences ($+4, +8, +16, +32$) double at every step—a guaranteed mathematical signature of a $\times 2 \pm k$ recurrence.
• Constant Multiplier with Negative Offset: In $2, 5, 14, 41, 122, ?$, each term follows $a_{n+1} = 3a_n - 1$ (or equivalently, first differences are $+3, +9, +27, +81$). The next difference is $+243$, giving $122 + 243 = 365$.
• Two-Step Operator Cycle: In $4, 8, 11, 22, 25, 50, 53, ?$, the operators alternate strictly between $\times 2$ and $+3$:

$$4 \xrightarrow{\times 2} 8 \xrightarrow{+3} 11 \xrightarrow{\times 2} 22 \xrightarrow{+3} 25 \xrightarrow{\times 2} 50 \xrightarrow{+3} 53 \xrightarrow{\times 2} 106$$

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3. Interleaved / Alternating Dual Series (İkili / Atlama Dizileri)

Whenever a sequence oscillates up and down or contains 7 or more given terms, suspect an interleaved dual series immediately. Two independent sub-sequences occupy the odd positions ($a_1, a_3, a_5, a_7, \dots$) and the even positions ($a_2, a_4, a_6, a_8, \dots$).

Worked Example 2: Interleaved Dual Series

Find the pair $(x, y)$ that continues the sequence: $5, 20, 8, 16, 11, 12, 14, 8, x, y$

1. Extract Odd-Indexed Terms ($a_1, a_3, a_5, a_7, a_9$):

$$5 \xrightarrow{+3} 8 \xrightarrow{+3} 11 \xrightarrow{+3} 14 \xrightarrow{+3} x = 17$$

1. Extract Even-Indexed Terms ($a_2, a_4, a_6, a_8, a_{10}$):

$$20 \xrightarrow{-4} 16 \xrightarrow{-4} 12 \xrightarrow{-4} 8 \xrightarrow{-4} y = 4$$

The missing pair is $(17, 4)$.

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4. Power, Prime, Factorial & Recursive Fibonacci-Type Series

Some sequences depend on canonical number-theoretic sets or multi-term recurrences rather than single-step arithmetic:

• Shifted Squares and Cubes: Memorize squares up to $20^2 = 400$ and cubes up to $10^3 = 1000$. A sequence like $0, 7, 26, 63, 124, ?$ is immediately recognizable as $n^3 - 1$ for $n = 1, 2, 3, 4, 5, 6$, yielding $6^3 - 1 = 215$. Similarly, $2, 5, 10, 17, 26, 37$ represents $n^2 + 1$.
• Prime-Indexed Shifts: The prime numbers ($2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, \dots$) often appear either as terms ($p_n \pm 1$) or as first differences ($\Delta_1 = +2, +3, +5, +7, +11, \dots$). Be careful not to confuse $+3, +5, +7$ (odd numbers, next $+9$) with $+2, +3, +5, +7$ (primes, next $+11$).
• Factorials: $1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120, 6! = 720$, where the step multipliers increase consecutively ($\times 2, \times 3, \times 4, \times 5, \times 6$).
• Fibonacci & Lucas Recursions: Each term is the sum (or product) of the two preceding terms:

$$a_n = a_{n-1} + a_{n-2}$$

For instance, in $3, 4, 7, 11, 18, 29, ?$, each term equals the sum of the two terms before it ($18 + 29 = 47$).

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5. Digit-Sum and Positional Number Series (Rakamları Toplamı Dizileri)

When first differences look erratic (for example, $+8, +7, +5, +10, +11$) and no polynomial or geometric rule fits, inspect the base-10 digits of each term. In a digit-sum recurrence, the step added to $a_n$ is the sum (or product) of the digits of $a_n$ itself:

$$a_{n+1} = a_n + S(a_n)$$

Worked Example 3: Self-Referential Digit-Sum Progression

Find the next term in: $17, 25, 32, 37, 47, 58, ?$

1. Check first differences: $+8, +7, +5, +10, +11$. These are neither monotonic nor periodic.
2. Compare each difference to the digits of the preceding term:
   • $17 \to 1 + 7 = 8 \implies 17 + 8 = 25$
   • $25 \to 2 + 5 = 7 \implies 25 + 7 = 32$
   • $32 \to 3 + 2 = 5 \implies 32 + 5 = 37$
   • $37 \to 3 + 7 = 10 \implies 37 + 10 = 47$
   • $47 \to 4 + 7 = 11 \implies 47 + 11 = 58$
3. Apply to $58$: The digit sum is $5 + 8 = 13$, so $? = 58 + 13 = 71$.

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Master Reference Table: TR-YÖS Number Series Archetypes

Checking What the Evidence Actually Determines

A short finite list does not determine a unique mathematical sequence unless the question restricts the rule. For the three terms 2, 4, 8, doubling gives a next term of 16, but the expression $a_n=n^2-n+2$ gives the same first three terms and then 14. Both fit the available observations. A fourth supplied term can reject one candidate: if it is 16, the quadratic rule fails. This explains why checking every supplied transition matters more than finding one attractive coincidence.

In an original multiple-choice pattern exercise, prefer a simple rule that consistently accounts for the whole display and the stated constraints. Record the operation and its scope: doubling every term is different from doubling only the odd-position subsequence. If two equally plausible rules lead to different offered answers, recheck the diagram, alternating positions and any explicit restrictions. Do not describe a pattern guess as a proof of uniqueness. This distinction also helps during review: a wrong answer can result from overlooking a second track rather than from performing the arithmetic incorrectly.

Review the Method

Non-standard sequences include shifted powers (n² ± 1, n³ ± 1), consecutive primes, factorials, Fibonacci-type recursions, and self-referential digit-sum progressions