What is Sum of n Terms of a GP? | Simple Explanation for Class 7

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What is the Sum of n Terms of a GP?

Grade Level:

Class 7

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

Definition

What is it?

The sum of n terms of a Geometric Progression (GP) is the total value you get when you add up a specific number of terms in a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. It helps us quickly find the total without adding each term individually.

Simple Example

Quick Example

Imagine you get Rs. 2 on Day 1, Rs. 4 on Day 2, Rs. 8 on Day 3, and so on, doubling each day. This is a GP. If you want to know how much money you have collected in total after 5 days, you would use the formula for the sum of n terms of a GP.

Worked Example

Step-by-Step

Let's find the sum of the first 4 terms of the GP: 3, 6, 12, 24, ...

Step 1: Identify the first term (a) and the common ratio (r).
Here, a = 3. To find r, divide the second term by the first: r = 6 / 3 = 2.

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Step 2: Identify the number of terms (n).
We need the sum of the first 4 terms, so n = 4.

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Step 3: Choose the correct formula. Since r (2) is greater than 1, we use the formula: S_n = a(r^n - 1) / (r - 1).

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Step 4: Substitute the values into the formula.
S_4 = 3 * (2^4 - 1) / (2 - 1)

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Step 5: Calculate the exponent.
2^4 = 2 * 2 * 2 * 2 = 16.

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Step 6: Continue solving.
S_4 = 3 * (16 - 1) / (1)
S_4 = 3 * (15) / 1
S_4 = 45

The sum of the first 4 terms of the GP is 45.

Why It Matters

Understanding GP sums is crucial for calculating compound interest on your savings, predicting population growth, or even figuring out how fast a computer algorithm processes data. It's used by bankers, scientists, and software engineers to solve real-world problems and make smart decisions.

Common Mistakes

MISTAKE: Using the formula for Arithmetic Progression (AP) instead of GP. | CORRECTION: Always check if the sequence has a common difference (AP) or a common ratio (GP) before applying the sum formula.

MISTAKE: Forgetting to put 'r^n' inside the bracket before subtracting 1, or calculating (r-1) before raising r to the power of n. | CORRECTION: Follow the order of operations (BODMAS/PEMDAS): calculate r^n first, then subtract 1, then multiply by 'a', and finally divide by (r-1).

MISTAKE: Using the wrong formula when r is less than 1. | CORRECTION: If r < 1, use S_n = a(1 - r^n) / (1 - r) to avoid negative numbers in the denominator and make calculations easier. Both formulas give the same result, but this one is simpler for r < 1.

Practice Questions

Try It Yourself

QUESTION: Find the sum of the first 3 terms of the GP: 5, 10, 20. | ANSWER: 35

QUESTION: What is the sum of the first 4 terms of a GP where the first term is 1 and the common ratio is 3? | ANSWER: 40

QUESTION: A new mobile app gains users such that on the first day it has 10 users, and each day the number of new users doubles. How many total users will the app have after 5 days? | ANSWER: 310 users

MCQ

Quick Quiz

Which formula is used to find the sum of n terms of a GP when the common ratio (r) is greater than 1?

S_n = n/2 * (2a + (n-1)d)

S_n = a(r^n - 1) / (r - 1)

S_n = a(1 - r^n) / (1 - r)

S_n = a + (n-1)d

The Correct Answer Is:

Option B is the correct formula for the sum of n terms of a GP when r > 1. Options A and D are for Arithmetic Progressions, and Option C is for GPs when r < 1.

Real World Connection

In the Real World

In finance, when you calculate Compound Interest on a Fixed Deposit (FD) or Recurring Deposit (RD) in an Indian bank, the interest earned each year forms a Geometric Progression. The total amount you receive at maturity is essentially the sum of terms of a GP. This concept helps banks and customers understand long-term savings growth.

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 | COMMON RATIO (r): The fixed number by which each term is multiplied to get the next term in a GP | FIRST TERM (a): The very first number in the sequence | NUMBER OF TERMS (n): How many terms are being added up

What's Next

What to Learn Next

Great job understanding the sum of n terms of a GP! Next, you should explore 'Sum to Infinity of a GP'. This builds on what you've learned and shows you how to find the sum of an endless GP under certain conditions, which is fascinating!