Number Series and Missing Terms

What you will learn

You will complete numerical sequences under a declared rule, show a checkable calculation and verify an internal blank from both sides. This lesson selects six useful families: constant difference, constant ratio, a fixed change in gaps, alternating subsequences, a stated two-term recurrence, and stated square/cube rules. They are teaching coverage, not a claim that every possible RRB series has one of these forms.

You need exact addition, subtraction, multiplication, integer division and small positive powers from the arithmetic foundation, plus the habit of checking every example. Revise fraction arithmetic only if you need support with it; the present numerical entries use exact integer values. You do not need to complete the square-root lesson before using small squares. Letter-series mastery is not a compulsory gate, though its position and gap rows are useful models.

A sequence is an ordered list of terms. Term position 4 means the fourth entry, not the number 4. A finite prefix does not, by itself, prove a unique continuation. In every assessed question here, the stem states the permitted rule family or a finite allowed-rule menu. Check the answer within that domain rather than inventing an exception to rescue an option.

1 Equal differences: add the same signed amount

For a constant-difference sequence, calculate next term minus current term. An increasing sequence can have a positive step and a decreasing sequence a negative step. Keep the sign. Test all available differences; two numbers alone do not check that the same change continues throughout the given list.

Worked example

Consecutive terms differ by the same number.

13, 20, ?, 34, 41

Answer: 27

The difference is +7. Both 20+7=27 and 34−7=27 agree.

Watch the mistake: Use the terms on both sides of a blank.

After filling the blank, the complete list is 13,20,27,34,41 and every gap is +7. This is a two-sided check: a proposed middle term must connect correctly to both neighbours. A value chosen only from the first half of the list may fail the second half.

2 Equal ratios: multiply by the same factor

If the question specifies a fixed multiplier, compare next term ÷ current term. Use ratios only where the current term is nonzero; dividing by zero is not a valid test. A multiplying rule need not make a sequence larger. Multiplying by 1/3 is the same as dividing by 3, so positive terms become smaller.

Worked example

Every term is multiplied by the same positive factor to obtain the next term.

324, 108, 36, ?, 4

Answer: 12

The factor is 1/3: 324÷3=108, 108÷3=36, 36÷3=12, 12÷3=4.

Watch the mistake: A multiplying rule can make values smaller; do not force constant subtraction.

The ratio is constant even though the differences are not. Do not force a subtraction rule merely because the sequence decreases. Conversely, do not multiply merely because a sequence increases: the stem's declared family and all the supplied terms must support that choice.

3 A rule for the gaps

When the gaps themselves change by a constant amount, write two rows:

• The first row contains the original terms
• The second row contains next term minus current term

Then compare neighbouring gaps. “The gaps increase by 3” means adding 3 to the previous gap, not adding 3 directly to the last original term. The next term requires two steps: find the new gap, then add it to the last term.

Worked example

The gaps between consecutive terms increase by a fixed amount.

5, 9, 16, 26, 39, ?

Answer: 55

Gaps 4,7,10,13 increase by 3; next gap is 16, so 39+16=55.

Watch the mistake: 13 is the last gap, not the next gap.

Check the complete difference row: 4,7,10,13,16. Its consecutive changes are 3,3,3,3. This explanation supports 55 under the specified gap family; it is not a proof about every imaginable formula that could pass through the finite data.

4 Separate subsequences by position

A question may specify two constant-difference sequences taking turns. Number all term positions before splitting them. Put positions 1,3,5,… in one row and positions 2,4,6,… in another. Do not sort the numbers by whether their values are odd or even.

Worked example

Odd and even term positions each form a constant-difference sequence.

7, 40, 12, 36, 17, 32, ?, 28

Answer: 22

Odd positions: 7,12,17,22 (+5). Even positions: 40,36,32,28 (−4).

Watch the mistake: Do not compare 17 directly with 32 to extend the odd-position sequence.

The missing seventh term is in the odd-position row, even though its value 22 is even. This is exactly why term position and term value must remain separate. Check the other row too: an appealing rule for the target row cannot excuse a contradiction in the supplied even row.

5 A stated rule using the previous two terms

Some sequences give a recurrence: a rule for making a new term from previous terms. You do not need that technical word to solve the question. Read how many earlier terms are used and keep the starting terms supplied by the question.

For the rule “add the previous two”, the first two terms are starting information. The rule begins at the third term. At each new step, move the two-term window forward; do not keep adding the first starting value forever.

Worked example

From the third term onward, each term equals the sum of the previous two terms.

4, 9, 13, 22, 35, ?

Answer: 57

4+9=13, 9+13=22, 13+22=35, so 22+35=57.

Watch the mistake: Add the two latest terms, not the first term and the latest term.

The exact rule matters more than recognising a famous sequence name. Different starting values can obey the same recurrence, and they produce different numerical lists.

6 Consecutive powers and a stated adjustment

A square is n²=n×n; a cube is n³=n×n×n. The base n and the resulting term are different quantities. A question may state that consecutive increasing positive bases are squared or cubed and then the same amount is added or subtracted. Track all three parts: base order, exponent, and adjustment.

Worked example

The terms are one more than the squares of consecutive increasing positive integers.

10, 17, 26, 37, 50, ?

Answer: 65

The bases are 3,4,5,6,7; the next is 8, and 8²+1=65.

Watch the mistake: 64 omits the stated extra one; 50 is not itself a square.

Write the supplied checks fully: 3²+1=10, 4²+1=17, 5²+1=26, 6²+1=37 and 7²+1=50. The next base is 8. If a question instead states a cube rule, use three factors, not two; do not infer the exponent merely because one entry happens to be both a square and a cube.

7 Choose a representation, then verify

Use the rule family stated in the question to choose the working row. There is no compulsory routine of trying squares first, then cubes, then multiplication for every problem.

• Constant difference: subtract adjacent terms
• Constant ratio: divide adjacent nonzero terms
• Changing gaps: make and extend the difference row
• Alternation: label and separate the two sets of positions
• Previous-two recurrence: check every term from the third onward
• Powers with adjustment: recover the positive bases and keep the stated direction

For an internal blank, reconstruct the full sequence and check both sides. If a proposed rule fails one given term, do not ignore that term. If more than one allowed rule yields different targets, the data are insufficient for that declared menu; choosing a favourite pattern does not remove the ambiguity.

Independent practice

Attempt all six before the explained key. This free learning practice is untimed, with +1 for correct and 0 for incorrect or unattempted, no negative marking and no pass cutoff. The settings are pedagogic, not the RRB exam's marking scheme.

1. Consecutive terms have one constant difference: 46, 39, ?, 25, 18. Find the missing term.

A. 31 B. 33 C. 32 D. 34

2. Each term is multiplied by the same positive factor to obtain the next: 5, 15, 45, ?, 405. What is the missing term?

A. 75 B. 135 C. 45 D. 405

3. The gaps between consecutive terms increase by the same fixed amount: 8, 13, 21, 32, ?. Find the next term.

A. 43 B. 45 C. 48 D. 46

4. Odd term positions and even term positions each form their own constant-difference sequence: 6, 51, 10, 46, 14, 41, 18, ?. Find the term at position 8.

A. 36 B. 37 C. 22 D. 46

5. From the third term onward, every term is the sum of the previous two: 6, 10, 16, 26, 42, ?. Find the next term.

A. 52 B. 58 C. 68 D. 84

6. Each term is one less than the cube of consecutive increasing positive integers: 7, 26, 63, 124, ?. Find the next term.

A. 216 B. 215 C. 124 D. 342

Explained key

1. C — 32

The step is −7 because 39−46=−7 and 18−25=−7. The blank is 39−7=32; checking forward gives 32−7=25. The completed differences are −7,−7,−7,−7. A gives −8 then −6 around the blank; B gives −6 then −8; D gives −5 then −9.

2. B — 135

15÷5=3 and 45÷15=3, so the multiplier is 3. The missing term is 45×3=135, and 135×3=405 confirms the other side. A adds the previous difference 30 instead of multiplying. C repeats 45. D skips the missing stage and copies the later term; 405×3 would not give the stated next value 405.

3. D — 46

The gaps are 5,8,11, increasing by 3 each time. The next gap is 14 and the next term is 32+14=46. A repeats the last gap 11. B uses 13, which increases the last gap by only 2. C uses 16, which increases it by 5. Only D continues the stated constant change of 3.

4. A — 36

Odd positions contain 6,10,14,18 with +4. Even positions contain 51,46,41,36 with −5. Position 8 belongs to the even row, so 41−5=36. B subtracts 4, borrowing the size of the odd-row step. C extends the odd row to 22 even though its next term would occupy position 9. D repeats the earlier even-row term 46 and reverses the direction of the last step.

5. C — 68

Check the given terms: 6+10=16, 10+16=26 and 16+26=42. The latest two are 26 and 42, so the next term is 68. A uses the old seed 10 with 42. B uses 16 with 42 and skips 26. D doubles 42 instead of adding the two different preceding terms.

6. B — 215

The supplied terms are 2³−1=7, 3³−1=26, 4³−1=63 and 5³−1=124. The next base is 6, so 6³−1=216−1=215. A omits the final subtraction. C repeats the previous term. D is 7³−1 and skips the required base 6.

Source and scope

RRB CEN 09/2025, §14.1, printed p.28, names alphabetical and number series, coding and decoding, and mathematical operations. The topic list is illustrative, not exhaustive. NCERT Ganita Prakash, Chapter 1, §§1.2–1.4, supports the basic pattern ideas. All explanations, examples and questions here are original teaching material. The selected subtypes and question counts are learning choices, not official topic weightage. These are not previous-year questions or a full reasoning course.

Analogy

Imagine following a written recipe for a row of numbered cards. “Add the same amount” is one recipe; “multiply by the same factor” is another. Two alternating rows use two recipes taking turns. A missing card must agree with the recipe before and after its position. Seeing a few cards does not identify every possible recipe, so the question must specify the permitted kind.

Quick reference

• Read the explicit rule family before choosing a method
• Term position and term value are different
• Difference = next − current; keep the sign
• Ratio = next ÷ current, only when current ≠ 0
• Constant change in gaps: extend the gap row, then find the term
• Alternation: split by term indices, not parity of values
• Sum-of-previous-two rule begins at term 3; retain both starting terms
• Powers: track consecutive positive bases, exponent and adjustment
• Internal blank: check both neighbouring steps and then the entire list
• A keyed answer is conditional on the declared rule domain