# Series – Reasoning Ability
(Jharkhand TET – Teacher Eligibility Test)
1. What is a “Series”?
A series is a list of numbers (or letters, words, shapes, etc.) that follows a fixed rule or pattern.
In reasoning tests you are asked to:
- Spot the rule.
- Find the next term(s).
- Sometimes locate a missing term or the (n^{th}) term.
2. Common Types of Numerical Series
| Type | Typical Pattern | Formula / Rule | Example |
|---|---|---|---|
| Arithmetic (AP) | Constant difference (d) between consecutive terms | (a_n = a_1 + (n-1)d) | 3, 7, 11, 15 … |
| Geometric (GP) | Constant ratio (r) between consecutive terms | (a_n = a_1 \times r^{,n-1}) | 2, 6, 18, 54 … |
| Fibonacci | Sum of the two preceding terms | (a_n = a_{n-1} + a_{n-2}) | 1, 1, 2, 3, 5, 8 … |
| Alternating | Two or more interleaved patterns | e.g., even terms follow AP, odd terms follow GP | 2, 4, 6, 8, 10 (AP) and 3, 9, 27, 81 (GP) |
| Difference Series | Differences themselves follow a simple pattern | e.g., differences increase by 2: 5, 7, 9 … | 1, 6, 13, 24 … |
| Square / Cube Series | Terms are perfect squares or cubes | (n^2) or (n^3) | 1, 4, 9, 16, 25 … |
| Factorial / Double‑factorial | (n!) or (n!!) | (n! = n \times (n-1)!) | 1, 2, 6, 24, 120 … |
| Prime / Composite | Prime or composite numbers in order | List of primes: 2, 3, 5, 7 … | |
| Alternating Sum/Pattern | Terms alternate in sign or magnitude | e.g., 5, –3, 2, –1, … |
3. Quick Tips for Identifying the Pattern
| Step | What to Look For |
|---|---|
| 1. Check the first few terms | Are they increasing/decreasing? |
| 2. Compute differences | Constant → AP; increasing/decreasing by a constant → difference series |
| 3. Compute ratios | Constant → GP |
| 4. Look for squares/cubes | 1, 4, 9, 16 … or 1, 8, 27, 64 … |
| 5. Test “sum of previous two” | Fibonacci or similar |
| 6. Notice alternation | Separate odd/even positions |
| 7. Check for negative signs | Alternating sign series |
| 8. Look for letters/words | Alphabetic positions, word lengths, etc. |
4. Worked Example 1
Series: 2, 5, 10, 17, 26, …
Goal: Find the next term.
| Step | Calculation |
|---|---|
| 1. Compute differences: | 5–2 = 3, 10–5 = 5, 17–10 = 7, 26–17 = 9 |
| 2. Observe differences: 3, 5, 7, 9 → increase by 2 each time. | |
| 3. Next difference = 9 + 2 = 11. | |
| 4. Next term = last term + next difference = 26 + 11 = 37. |
Answer: 37
5. Worked Example 2
Series: 3, 9, 27, 81, …
Goal: Find the 5th term.
| Step | Calculation |
|---|---|
| 1. Recognize ratio: 9/3 = 3, 27/9 = 3, 81/27 = 3 → Geometric with (r=3). | |
| 2. Use formula (a_n = a_1 \times r^{,n-1}). | |
| 3. (a_5 = 3 \times 3^{,4} = 3 \times 81 = 243). |
Answer: 243
6. Worked Example 3 – Mixed Pattern
Series: 1, 4, 9, 16, 25, 36, …
Goal: Identify the pattern and find the 8th term.
| Step | Calculation |
|---|---|
| 1. Notice that each term is a perfect square: (1^2, 2^2, 3^2, …). | |
| 2. Formula (a_n = n^2). | |
| 3. 8th term (a_8 = 8^2 = 64). |
Answer: 64
7. Worked Example 4 – Alternating Sign Series
Series: 1, –1, 2, –2, 3, …
Goal: Find the 7th term.
| Step | Calculation |
|---|---|
| 1. Separate odd/even positions. | |
| 2. Odd positions: 1, 2, 3 → increasing by 1. | |
| 3. Even positions: –1, –2 → decreasing by 1. | |
| 4. 7th term is odd position → value = 4. | |
| 5. Since odd positions are positive → 4. |
Answer: 4
8. Quick Revision Summary
| Pattern | Key Feature | Formula (if applicable) | Quick Check |
|---|---|---|---|
| Arithmetic | Constant difference (d) | (a_n = a_1 + (n-1)d |