Series

Reasoning Ability – Series

Prepared for Manipur TET (Teacher Eligibility Test)

1. What is a “Series”?

A series is a set of numbers arranged in a particular order, usually following a rule or pattern.
In Reasoning, we often need to:

• Identify the rule that generates the series.
• Predict the next term(s).
• Find the missing term(s).
• Compute the sum of a given number of terms.

2. Common Types of Series

Tip: Always check the difference or ratio first; if neither is constant, look for patterns in squares, factorials, or Fibonacci-type sums.

3. Key Points to Remember

1. Identify the Rule

   • Compute first differences (Δa = a₂ - a₁).
   • Compute first ratios (a₂ / a₁).
   • If neither is constant, check for other patterns (squares, cubes, Fibonacci, etc.).

2. Check for Alternation

   • Even–odd positions may follow separate patterns.
   • Signs may alternate: +, –, +, –.

3. Use Formulae

   • For AP: Sum of first n terms = n/2 × (2a₁ + (n‑1)d).
   • For GP: Sum of first n terms = a₁ (1 - rⁿ) / (1 - r) (if r ≠ 1).

4. Look for Hidden Patterns

   • Differences of differences (second-order differences) can reveal a quadratic pattern.
   • Ratios of differences can reveal a geometric pattern.

5. Check the End of the Series

   • The last term may give clues about the pattern (e.g., 2, 4, 8, 16 – clearly powers of 2).

6. Practice with Sample Questions

   • The more you practice, the quicker you’ll spot the pattern.

4. Worked Example

Question

The series 3, 9, 27, 81, … continues. What is the 5th term and what is the sum of the first 5 terms?

Step‑by‑Step Solution

1. Identify the Pattern

   • Ratio of consecutive terms:
     • 9 ÷ 3 = 3
     • 27 ÷ 9 = 3
     • 81 ÷ 27 = 3
   • Since the ratio r = 3 is constant, it’s a Geometric Progression (GP).

2. Find the 5th Term

   • Formula for the nth term:
     [ a_n = a_1 \times r^{,n-1} ]
   • Plugging values:
     [ a_5 = 3 \times 3^{,5-1} = 3 \times 3^4 = 3 \times 81 = 243 ]

3. Sum of First 5 Terms

   • Sum formula for GP (when r ≠ 1):
     [ S_n = a_1 \frac{r^n - 1}{r - 1} ]
   • Plugging values:
     [ S_5 = 3 \frac{3^5 - 1}{3 - 1} = 3 \frac{243 - 1}{2} = 3 \frac{242}{2} = 3 \times 121 = 363 ]

Answers

• 5th Term: 243
• Sum of first 5 terms: 363

Quick Check:

• Terms: 3, 9, 27, 81, 243 (each multiplied by 3).
• Sum: 3 + 9 + 27 + 81 + 243 = 363. ✔️

5. Quick Revision Summary