How to Find the General Term of a Sequence: Step-by-Step Guide for CAT/XAT

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📈 How to Find the General Term of a Sequence (T_n) – Step-by-Step Guide for CAT/XAT

One of the most frequently asked concepts in CAT/XAT and other MBA entrance exams is identifying the general term of a sequence. If you’re given a list of numbers like:

2, 5, 10, 17, 26, ...

Can you predict the general term T_n for the nth position? Let’s learn a foolproof method to tackle these questions.

🔢 Step 1: Understand the Problem – What Is T_n?

The general term \( T_n \) represents the value at the nth position in a sequence. Finding a formula for \( T_n \) allows you to find the 50th, 100th, or even the 1000th term without writing out every single value.

There are common types of sequences:

Arithmetic: Constant difference → Linear → Degree 1
Quadratic: Constant second difference → Degree 2
Cubic: Constant third difference → Degree 3
Quartic: Constant fourth difference → Degree 4

Above: Difference table for the sequence 2, 5, 10, 17, 26,...

🔍 Step 2: Use the Difference Table

Take this sequence:

6, 13, 24, 39, 58

First, compute the first-level differences:

13 – 6 = 7
24 – 13 = 11
39 – 24 = 15
58 – 39 = 19

Now the second differences:

11 – 7 = 4
15 – 11 = 4
19 – 15 = 4

✅ Second differences are constant ⇒ Degree 2 ⇒ General term is a quadratic:

T_n = a·n² + b·n + c

✍️ Step 3: Use Term Substitution to Find a, b, c

Use the first 3 terms to form equations:

For T₁ = 6 → \( a + b + c = 6 \)
For T₂ = 13 → \( 4a + 2b + c = 13 \)
For T₃ = 24 → \( 9a + 3b + c = 24 \)

Solve this system (you can use elimination or matrix method) and get:

T_n = 2n² + 4

Check: T₁ = 2(1)² + 4 = 6 ✅, T₂ = 2(4) + 4 = 13 ✅, and so on.

🎯 Step 4: Answer Typical Questions

Once you have the general term, you can easily answer:

🔹 What is the 10th term? → Just plug n = 10
🔹 What is the next term? → Plug n = last index + 1
🔹 What is the degree of the summation? → If T_n is degree d, sum is degree d + 1

Example: What is the 10th term of T_n = 2n² + 4?

Put n = 10 → T₁₀ = 2(100) + 4 = 204

📚 Another Example: Cubic Case

Given: 2, 15, 52, 129, 262

Compute differences till 3rd level:

1st diff: 13, 37, 77, 133
2nd diff: 24, 40, 56
3rd diff: 16, 16

✅ Third differences are constant ⇒ Degree = 3 Assume:

T_n = a·n³ + b·n² + c·n + d

Use first 4 terms to solve for a, b, c, d → You get:

T_n = n³ + n² + 2

Now you can compute any term instantly!

📌 Tips for CAT/XAT

✅ Always check how many differences it takes to get constant values → that’s your degree
✅ Use known values to form equations and solve step-by-step
✅ Don't assume arithmetic unless first differences are equal!

🧠 Challenge Question

Given the sequence: 5, 14, 29, 50, 77

Find the general term
Find the 6th and 10th term
What is the degree of summation?

Click to reveal answer

\( T_n = 3n^2 + 2 \)

6th term = 110
10th term = 302
Degree of summation: 3

🚀 Final Thoughts

This is a high-yield concept for CAT/XAT Quant. Mastering this skill not only helps in sequence questions but also strengthens your algebraic thinking. Bookmark this guide and practice 3–4 sequences daily for mastery!

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