What is the nth Term of a GP?

S3-SA1-0158

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The 'nth term' of a Geometric Progression (GP) is a formula that helps you find any term in the sequence without having to list all the terms before it. It's like a shortcut to find the 5th, 10th, or even 100th number in a special pattern where each number is found by multiplying the previous one by a fixed value.

Simple Example

Quick Example

Imagine you start with 2 ladoos on Day 1. Each day, you double the number of ladoos. So, Day 1 has 2, Day 2 has 4, Day 3 has 8, and so on. If you want to know how many ladoos you'd have on Day 5 without counting, the nth term formula helps you find it directly.

Worked Example

Step-by-Step

Let's find the 6th term of the GP: 3, 6, 12, 24, ...

Step 1: Identify the first term (a). Here, a = 3.
---Step 2: Find the common ratio (r). Divide any term by its previous term. r = 6/3 = 2.
---Step 3: Identify 'n', the term number we want to find. We want the 6th term, so n = 6.
---Step 4: Write down the formula for the nth term of a GP: a_n = a * r^(n-1).
---Step 5: Substitute the values into the formula: a_6 = 3 * 2^(6-1).
---Step 6: Calculate the exponent: a_6 = 3 * 2^5.
---Step 7: Calculate the power: 2^5 = 2 * 2 * 2 * 2 * 2 = 32.
---Step 8: Multiply to get the final term: a_6 = 3 * 32 = 96.
So, the 6th term of the GP is 96.

Why It Matters

Understanding the nth term of a GP helps in fields like Data Science to model population growth or decay, and in Finance to calculate compound interest on your savings. Engineers use it to design systems where growth or decrease happens in a fixed ratio, like how signals weaken over distance.

Common Mistakes

MISTAKE: Forgetting to subtract 1 from 'n' in the exponent (r^n instead of r^(n-1)). | CORRECTION: Always remember the formula is a * r^(n-1). The exponent is always one less than the term number you're looking for.

MISTAKE: Confusing GP with AP (Arithmetic Progression) and using the wrong formula. | CORRECTION: Remember that in GP, you multiply by a common ratio (r), while in AP, you add a common difference (d). The formulas are very different!

MISTAKE: Incorrectly calculating powers, especially for larger numbers (e.g., saying 2^3 is 6 instead of 8). | CORRECTION: Always calculate powers carefully by multiplying the base number by itself the correct number of times.

Practice Questions

Try It Yourself

QUESTION: Find the 4th term of the GP: 5, 10, 20, ... | ANSWER: 40

QUESTION: What is the 5th term of a GP where the first term is 2 and the common ratio is 3? | ANSWER: 162

QUESTION: A GP has its first term as 100 and a common ratio of 1/2. What is the 7th term of this GP? | ANSWER: 100 * (1/2)^6 = 100 * (1/64) = 100/64 = 25/16

MCQ

Quick Quiz

What is the 3rd term of a GP with first term 4 and common ratio -2?

-8

-16

The Correct Answer Is:

The formula is a_n = a * r^(n-1). For the 3rd term, a_3 = 4 * (-2)^(3-1) = 4 * (-2)^2 = 4 * 4 = 16. Options B and D are wrong due to sign errors, and A is incorrect calculation.

Real World Connection

In the Real World

When you invest money in a fixed deposit (FD) at a bank, the interest often compounds, meaning it grows on the previous year's interest. This growth can be modeled using a Geometric Progression. Similarly, how a virus spreads in the initial stages can follow a GP pattern, helping health officials predict its reach.

Key Vocabulary

Key Terms

GEOMETRIC PROGRESSION (GP): A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number | FIRST TERM (a): The starting number in a GP | COMMON RATIO (r): The fixed number by which each term is multiplied to get the next term | NTH TERM: Any specific term in the sequence at position 'n'

What's Next

What to Learn Next

Great job learning about the nth term of a GP! Next, you can explore the 'Sum of n Terms of a GP'. This will teach you how to quickly add up all the numbers in a GP, which is super useful for calculating total earnings from compound interest or total growth over time.