14.2 Number Series: Differences, Powers & Missing Terms

Number Series & Missing Term Prediction

Number series questions require candidates to identify the generative rule linking consecutive terms in a sequence and predict the missing term ($?$) or identify the aberrant wrong term.

1. Constant Difference Series (Arithmetic Progression)

In an arithmetic series, consecutive terms differ by a constant value $d$:

$$T_n = T_1 + (n - 1)d$$

Example: $14, 23, 32, 41, 50, \dots$ ($d = +9$).

2. Difference of Differences (Second-Order and Third-Order Differences)

When first differences between consecutive terms are not constant, compute successive difference tiers:

Series Terms (Tn):       T1         T2         T3         T4         T5
                          \       /  \       /  \       /  \       /
First Difference (Δ1):        d1         d2         d3         d4
                               \       /  \       /  \       /
Second Difference (Δ2):            D1         D2         D3   (Constant = c)

• Second-Order Difference Rule: If the second difference $\Delta_2$ is constant ($c$), the sequence is governed by a quadratic polynomial in $n$:

$$T_n = an^2 + bn + c_0$$

The coefficient of $n^2$ is strictly half of the second constant difference: $a = \frac{\Delta_2}{2}$.

• Third-Order Difference Rule: If the third difference $\Delta_3$ is constant ($k$), the sequence is cubic:

$$T_n = an^3 + bn^2 + cn + d_0 \quad \text{where } a = \frac{\Delta_3}{6}$$

3. Fibonacci and Special Sequences

• Fibonacci Series: Each term is the sum of the preceding two terms:

$$T_n = T_{n-1} + T_{n-2} \quad (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, \dots)$$

• Triangular Numbers: The sum of the first $n$ natural numbers:

$$T_n = \frac{n(n + 1)}{2} \quad (1, 3, 6, 10, 15, 21, 28, 36, \dots)$$

Worked Example 1: Second-Order Difference of Differences (Missing Term)

Problem: Determine the missing term in the sequence: $5, 14, 29, 50, 77, ?$.

• Step 1: Compute First-Order Differences ($\Delta_1$):
  • $14 - 5 = 9$
  • $29 - 14 = 15$
  • $50 - 29 = 21$
  • $77 - 50 = 27$
  • The first difference sequence is: $9, 15, 21, 27$.
• Step 2: Compute Second-Order Differences ($\Delta_2$):
  • $15 - 9 = 6$
  • $21 - 15 = 6$
  • $27 - 21 = 6$
  • The second difference is strictly constant: $\Delta_2 = 6$.
• Step 3: Extrapolate Next Differences:
  • Next first-order difference: $27 + 6 = 33$.
• Step 4: Compute Missing Term:
  • Missing term $= 77 + 33 = 110$.

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Choose a simple rule and test every transition

A finite list can fit many mathematical rules. In aptitude exercises, prefer the simplest consistent pattern supported by all supplied terms and the answer choices. A rule that explains the first two terms but fails the fourth is not a solution. State the pattern before calculating the missing term. This separates recognition from arithmetic and makes your answer checkable.

Start by comparing consecutive differences. For 8, 13, 18, 23, each difference is 5, so the next term is 28. For 2, 6, 12, 20, 30, the first differences are 4, 6, 8, 10. Their constant increase is 2; extend to 12 and obtain 42. The second example is not an arithmetic progression even though its differences form one.

If division gives a constant ratio instead, test multiplication. The series 3, 9, 27, 81 multiplies by 3. The next term is 243. Do not substitute an additive rule simply because the absolute increases grow. If zeros or negative numbers occur, division may be unavailable or misleading; use the stated operations and verify signs explicitly.

Square and cube recognition can save time, but verify offsets. The list 2, 5, 10, 17, 26 fits consecutive squares plus one. Its next term is 37. It does not follow the square sequence itself. Likewise, 0, 7, 26, 63 fits consecutive positive integer cubes minus one. Writing the generating rule prevents an off-by-one answer.

For a missing internal term, work from both sides. In 6, 11, ?, 21, 26, a constant difference of 5 gives 16 and agrees with both neighboring terms. A choice that fits only the left side should be rejected. If two plausible rules remain consistent and the question gives no further restriction, acknowledge that finite sequences are not mathematically unique rather than claiming a theorem about the examiner's intended rule.

Use consecutive differences as an independent check

For the list 4, 9, 16, 25, 36, the next term is 49 under the consecutive-square model beginning with 2 squared. The observed gaps 5, 7, 9 and 11 independently predict the next gap 13. Both descriptions give 49. Identifying a square model beginning with 1 squared would assign the wrong index even though the values look familiar.

Now place the blank inside the list: 4, 9, ?, 25, 36. A candidate answer 16 creates differences 5, 7, 9 and 11. A candidate answer 18 creates differences 5, 9, 7 and 11, breaking the constant increase. Verify the blank from both directions. For a later predicted term, count the number of transitions from the last known value; the next term is one transition away, not two.