Number Series questions with answers and concepts

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162 practice questionsPage updated 11 Nov 2026Reviewed by AISEA Editorial DeskPractice questions

Practise this concept

Number series questions ask you to identify the rule connecting successive terms and then use it to find a missing or next term. The rule may involve a constant difference, a constant ratio, powers, a recurrence, alternating positions or successive differences. A finite list can fit many invented rules, so prefer the simplest consistent rule that explains every term and fits the answer choices.

Key formulas and rules

5 rules

Arithmetic sequence: $a_{n} = a_{1} + (n - 1) d$ , where $d$ is the constant difference.

Geometric sequence: $a_{n} = a_{1} r^{n - 1}$ , where $r$ is the constant non-zero ratio.

Triangular numbers: $T_{n} = \frac{n ( n + 1 )}{2}$ .

Forward difference: $Δ a_{n} = a_{n + 1} - a_{n}$ ; second difference: $Δ^{2} a_{n} = Δ a_{n + 1} - Δ a_{n}$ .

Fibonacci recurrence: $F_{n} = F_{n - 1} + F_{n - 2}$ with the starting values supplied by the sequence.

Key concepts

Arithmetic progressions

Subtract consecutive terms. If the same difference appears throughout, the series is arithmetic. The difference may be positive or negative, and it may be fractional. For example, 18, 13, 8, 3 has d = -5, so the next term is -2.

Geometric progressions

Divide each term by the preceding term when division is defined. A constant ratio identifies a geometric pattern, such as 3, 6, 12, 24 with $r = 2$ . A decreasing series may have a fractional ratio, such as 81, 27, 9, 3 with $r = 1/3$ .

Powers and familiar sequences

Check common structured families after testing simple differences and ratios. Useful candidates include squares, cubes, primes, multiples, triangular numbers and factorials. Write the index beside each term, because a pattern such as n squared plus 1 becomes easier to see when the position is explicit.

Difference tables

If first differences are not constant, calculate differences again. A constant second difference often indicates a quadratic rule; a constant third difference can indicate a cubic rule. Use this method only after checking simpler patterns, and verify the derived next difference before selecting an answer.

Recurrence and alternating rules

Some series use previous terms, as in Fibonacci-type addition, or alternate operations, such as divide by 2 then multiply by 3. For alternating rules, separate odd-position and even-position terms and analyse each subsequence independently. Do not force one rule across positions that clearly behave differently.

Reverse operations and mixed patterns

A series may alternate addition and subtraction, multiply by a changing number, or combine an operation with a changing increment. Trace the operations between every pair: +3, +5, +7 is different from multiply by 2 and add 1. State the operation sequence before extending it.

Validate the candidate rule

A pattern is useful only when it explains all supplied terms. Check the rule from the first pair to the last, calculate the missing term independently, and compare it with the options. If two rules appear possible, choose the one with fewer assumptions and the most regular operation.

Check the idea while it is fresh.Open a focused exercise without losing this article.Practise this concept

Worked examples

Example 1

Find the next number in the sequence: $4, 9, 14, 19, ?$

Solution

Step 1: Find the common first difference.

9 - 4 = 5, 14 - 9 = 5, 19 - 14 = 5

This is an arithmetic progression with common difference $d = 5$ .

Step 2: Calculate the next term.

19 + 5 = 24

Answer: 24.

Example 2

Find the missing term in the geometric progression: $3, 6, 12, 24, ?, 96$ .

Solution

Step 1: Determine the common ratio.

r = \frac{6}{3} = \frac{12}{6} = 2

Step 2: Compute the missing term.

24 \times 2 = 48

Check: $48 \times 2 = 96$ (consistent).

Answer: 48.

Example 3

Find the missing number in the sequence: $5, 10, 17, 26, ?, 50$ .

Solution

Step 1: Express terms in the form $(n + 1)^{2} + 1$ .
- For $n = 1$ : $(1 + 1)^{2} + 1 = 4 + 1 = 5$
- For $n = 2$ : $(2 + 1)^{2} + 1 = 9 + 1 = 10$
- For $n = 3$ : $(3 + 1)^{2} + 1 = 16 + 1 = 17$
- For $n = 4$ : $(4 + 1)^{2} + 1 = 25 + 1 = 26$

Step 2: Calculate for $n = 5$ .

(5 + 1)^{2} + 1 = 6^{2} + 1 = 36 + 1 = 37

Check for $n = 6$ : $7^{2} + 1 = 50$ .

Answer: 37.

Example 4

Find the next number in the series: $3, 8, 15, 24, 35, ?$

Solution

Step 1: Compute first differences.

8 - 3 = 5, 15 - 8 = 7, 24 - 15 = 9, 35 - 24 = 11

Differences increase in steps of $+ 2$ .

Step 2: Next difference and term.

Next difference = 11 + 2 = 13

Next term = 35 + 13 = 48

Alternative check: $n^{2} + 2 n$ for $n = 1, 2, 3, 4, 5 ⟹ 6^{2} + 2 (6) = 36 + 12 = 48$ .

Answer: 48.

Example 5

Find the next term in the alternating sequence: $2, 5, 4, 10, 6, 15, ?$

Solution

Step 1: Partition into two interwoven sequences.
- Odd positions ( $1, 3, 5, 7$ ): $2, 4, 6, ? ⟹$ arithmetic sequence $+ 2$ .
- Even positions ( $2, 4, 6$ ): $5, 10, 15 ⟹$ arithmetic sequence $+ 5$ .

Step 2: Next term is at position 7 (odd position).

6 + 2 = 8

Answer: 8.

Example 6

Find the next term: $3, 5, 8, 13, 21, ?$

Solution

Step 1: Identify the additive recurrence ( $T_{n} = T_{n - 1} + T_{n - 2}$ ).

3 + 5 = 8, 5 + 8 = 13, 8 + 13 = 21

Step 2: Add the last two terms.

13 + 21 = 34

Answer: 34.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

What comes next: 1, 4, 10, 19, 31, ?

Answer: C. 46

Explanation

Step 1: First differences: 3, 6, 9, 12.
Step 2: Second difference pattern: +3 each time.
Step 3: Next first difference = 12+3 = 15.
Step 4: Next term = 31+15 = 46.
Answer: 46

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

What comes next: 4, 9, 25, 49, 121, ?

Answer: B. 169

Explanation

Step 1: These are squares of consecutive prime numbers.
Step 2: $2^{2} = 4$ , $3^{2} = 9$ , $5^{2} = 25$ , $7^{2} = 49$ , $1 1^{2} = 121$ .
Step 3: Next = $1 3^{2}$ = 169.
Answer: 169

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

What comes next: 0, 2, 6, 12, 20, ?

Answer: B. 30

Explanation

Step 1: Pattern: $n^{2} - n$ where n starts from 1.
Step 2: $1^{2} - 1 = 0$ , $2^{2} - 2 = 2$ , $3^{2} - 3 = 6$ , $4^{2} - 4 = 12$ , $5^{2} - 5 = 20$ .
Step 3: Next = $6^{2} - 6$ = 30.
Answer: 30

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

Start with consecutive differences, then ratios, before testing more elaborate rules.
Write term positions above the sequence to spot squares, cubes, triangular numbers and other index-based patterns.
For a changing difference, calculate first and second differences in a small table.
Split alternating series into odd and even positions instead of mixing two independent patterns.
Trace every operation between terms. A rule that works for the first three terms but fails later is not enough.
Use the answer choices as a verification tool, not as a reason to invent an irregular pattern.
Remember that a finite sequence may have multiple mathematical fits; exam patterns normally favour the simplest regular rule.

Sources and review notes

Reviewed by AISEA Editorial Desk on 1 Sep 2026.

Arithmetic, geometric, recurrence and finite-difference rules were checked against the cited sequence references.
Worked examples were recalculated term by term, including the shifted square rule and alternating subsequences.
The page explicitly warns that finite sequences can admit multiple fits and directs learners to prefer the simplest consistent rule.

Arithmetic ProgressionWolfram MathWorld · Checked 1 Sep 2026Supports: constant-difference arithmetic sequences and their common-difference notation
Geometric SequenceWolfram MathWorld · Checked 1 Sep 2026Supports: constant-ratio geometric sequences
Forward DifferenceWolfram MathWorld · Checked 1 Sep 2026Supports: first and higher finite differences for analysing sequence patterns
Fibonacci NumberWolfram MathWorld · Checked 1 Sep 2026Supports: the Fibonacci recurrence and starting-value convention

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Number Series questions with answers and concepts

162 practice questionsPage updated 11 Nov 2026Reviewed by AISEA Editorial DeskPractice questions

Practise this concept

Key formulas and rules

5 rules

Arithmetic sequence: $a_{n} = a_{1} + (n - 1) d$ , where $d$ is the constant difference.

Geometric sequence: $a_{n} = a_{1} r^{n - 1}$ , where $r$ is the constant non-zero ratio.

Triangular numbers: $T_{n} = \frac{n ( n + 1 )}{2}$ .

Forward difference: $Δ a_{n} = a_{n + 1} - a_{n}$ ; second difference: $Δ^{2} a_{n} = Δ a_{n + 1} - Δ a_{n}$ .

Fibonacci recurrence: $F_{n} = F_{n - 1} + F_{n - 2}$ with the starting values supplied by the sequence.

Key concepts

Arithmetic progressions

Geometric progressions

Powers and familiar sequences

Difference tables

Recurrence and alternating rules

Reverse operations and mixed patterns

Validate the candidate rule

Check the idea while it is fresh.Open a focused exercise without losing this article.Practise this concept

Worked examples

Example 1

Find the next number in the sequence: $4, 9, 14, 19, ?$

Solution

Step 1: Find the common first difference.

9 - 4 = 5, 14 - 9 = 5, 19 - 14 = 5

This is an arithmetic progression with common difference $d = 5$ .

Step 2: Calculate the next term.

19 + 5 = 24

Answer: 24.

Example 2

Find the missing term in the geometric progression: $3, 6, 12, 24, ?, 96$ .

Solution

Step 1: Determine the common ratio.

r = \frac{6}{3} = \frac{12}{6} = 2

Step 2: Compute the missing term.

24 \times 2 = 48

Check: $48 \times 2 = 96$ (consistent).

Answer: 48.

Example 3

Find the missing number in the sequence: $5, 10, 17, 26, ?, 50$ .

Solution

Step 2: Calculate for $n = 5$ .

(5 + 1)^{2} + 1 = 6^{2} + 1 = 36 + 1 = 37

Check for $n = 6$ : $7^{2} + 1 = 50$ .

Answer: 37.

Example 4

Find the next number in the series: $3, 8, 15, 24, 35, ?$

Solution

Step 1: Compute first differences.

8 - 3 = 5, 15 - 8 = 7, 24 - 15 = 9, 35 - 24 = 11

Differences increase in steps of $+ 2$ .

Step 2: Next difference and term.

Next difference = 11 + 2 = 13

Next term = 35 + 13 = 48

Alternative check: $n^{2} + 2 n$ for $n = 1, 2, 3, 4, 5 ⟹ 6^{2} + 2 (6) = 36 + 12 = 48$ .

Answer: 48.

Example 5

Find the next term in the alternating sequence: $2, 5, 4, 10, 6, 15, ?$

Solution

Step 2: Next term is at position 7 (odd position).

6 + 2 = 8

Answer: 8.

Example 6

Find the next term: $3, 5, 8, 13, 21, ?$

Solution

Step 1: Identify the additive recurrence ( $T_{n} = T_{n - 1} + T_{n - 2}$ ).

3 + 5 = 8, 5 + 8 = 13, 8 + 13 = 21

Step 2: Add the last two terms.

13 + 21 = 34

Answer: 34.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

What comes next: 1, 4, 10, 19, 31, ?

Answer: C. 46

Explanation

Step 1: First differences: 3, 6, 9, 12.
Step 2: Second difference pattern: +3 each time.
Step 3: Next first difference = 12+3 = 15.
Step 4: Next term = 31+15 = 46.
Answer: 46

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

What comes next: 4, 9, 25, 49, 121, ?

Answer: B. 169

Explanation

Step 1: These are squares of consecutive prime numbers.
Step 2: $2^{2} = 4$ , $3^{2} = 9$ , $5^{2} = 25$ , $7^{2} = 49$ , $1 1^{2} = 121$ .
Step 3: Next = $1 3^{2}$ = 169.
Answer: 169

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

What comes next: 0, 2, 6, 12, 20, ?

Answer: B. 30

Explanation

Step 1: Pattern: $n^{2} - n$ where n starts from 1.
Step 2: $1^{2} - 1 = 0$ , $2^{2} - 2 = 2$ , $3^{2} - 3 = 6$ , $4^{2} - 4 = 12$ , $5^{2} - 5 = 20$ .
Step 3: Next = $6^{2} - 6$ = 30.
Answer: 30

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

Start with consecutive differences, then ratios, before testing more elaborate rules.
Write term positions above the sequence to spot squares, cubes, triangular numbers and other index-based patterns.
For a changing difference, calculate first and second differences in a small table.
Split alternating series into odd and even positions instead of mixing two independent patterns.
Trace every operation between terms. A rule that works for the first three terms but fails later is not enough.
Use the answer choices as a verification tool, not as a reason to invent an irregular pattern.
Remember that a finite sequence may have multiple mathematical fits; exam patterns normally favour the simplest regular rule.

Sources and review notes

Reviewed by AISEA Editorial Desk on 1 Sep 2026.

Arithmetic, geometric, recurrence and finite-difference rules were checked against the cited sequence references.
Worked examples were recalculated term by term, including the shifted square rule and alternating subsequences.
The page explicitly warns that finite sequences can admit multiple fits and directs learners to prefer the simplest consistent rule.

Arithmetic ProgressionWolfram MathWorld · Checked 1 Sep 2026Supports: constant-difference arithmetic sequences and their common-difference notation
Geometric SequenceWolfram MathWorld · Checked 1 Sep 2026Supports: constant-ratio geometric sequences
Forward DifferenceWolfram MathWorld · Checked 1 Sep 2026Supports: first and higher finite differences for analysing sequence patterns
Fibonacci NumberWolfram MathWorld · Checked 1 Sep 2026Supports: the Fibonacci recurrence and starting-value convention

Ready to test your understanding?

Work through 162 questions with explanations after each answer.

Start practice

Next topic: Letter and Symbol Series

Arithmetic ProgressionWolfram MathWorld · Checked 1 Sep 2026Supports: constant-difference arithmetic sequences and their common-difference notation

Geometric SequenceWolfram MathWorld · Checked 1 Sep 2026Supports: constant-ratio geometric sequences

Forward DifferenceWolfram MathWorld · Checked 1 Sep 2026Supports: first and higher finite differences for analysing sequence patterns

Fibonacci NumberWolfram MathWorld · Checked 1 Sep 2026Supports: the Fibonacci recurrence and starting-value convention

Key formulas and rules

Key concepts

Arithmetic progressions

Geometric progressions

Powers and familiar sequences

Difference tables

Recurrence and alternating rules

Reverse operations and mixed patterns

Validate the candidate rule

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Example 5

Solution

Example 6

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Sources and review notes

Ready to test your understanding?

Keep building the same skill family.

Key formulas and rules

Key concepts

Arithmetic progressions

Geometric progressions

Powers and familiar sequences

Difference tables

Recurrence and alternating rules

Reverse operations and mixed patterns

Validate the candidate rule

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Example 5

Solution

Example 6

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Sources and review notes

Ready to test your understanding?

Keep building the same skill family.