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What is the Sum of n terms of a Geometric Progression?
Grade Level:
Class 6
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. Think of it as adding up numbers that grow or shrink by multiplying by the same number each time.
Simple Example
Quick Example
Imagine you start with 2 ladoos. Every day, you get twice the number of ladoos you got the previous day. So, Day 1: 2 ladoos, Day 2: 4 ladoos, Day 3: 8 ladoos. If you want to know the total ladoos you have after 3 days, you add them up: 2 + 4 + 8 = 14 ladoos. This total is the sum of 3 terms of this GP.
Worked Example
Step-by-Step
Let's find the sum of the first 4 terms of a GP where the first term (a) is 3 and the common ratio (r) is 2.
Step 1: Write down the first term (a) = 3.
---Step 2: Find the second term: a * r = 3 * 2 = 6.
---Step 3: Find the third term: 6 * r = 6 * 2 = 12.
---Step 4: Find the fourth term: 12 * r = 12 * 2 = 24.
---Step 5: Add all the terms together: Sum = 3 + 6 + 12 + 24.
---Step 6: Calculate the sum: Sum = 45.
So, the sum of the first 4 terms of this GP is 45.
Why It Matters
Understanding sums of GPs helps in many fields! In finance, it helps calculate how much your savings grow over time with compound interest. In computer science, it's used in designing efficient algorithms. Even scientists use it to model population growth or the decay of radioactive materials.
Common Mistakes
MISTAKE: Forgetting to multiply the first term by the common ratio to get the second term. | CORRECTION: Always remember that the common ratio (r) is used to find the NEXT term from the PREVIOUS term, starting from the second term.
MISTAKE: Adding the common ratio instead of multiplying it to find the next term. | CORRECTION: A Geometric Progression involves MULTIPLICATION by a common ratio, not addition. If you add, it becomes an Arithmetic Progression.
MISTAKE: Missing a term when calculating the sum for 'n' terms. For example, calculating sum of 3 terms when asked for 4. | CORRECTION: Carefully count the number of terms you are asked to sum and make sure you generate exactly that many terms before adding.
Practice Questions
Try It Yourself
QUESTION: What is the sum of the first 3 terms of a GP if the first term is 5 and the common ratio is 2? | ANSWER: 35
QUESTION: A plant grows such that its height doubles every week. If it starts at 3 cm, what is its total height after 4 weeks (sum of heights at end of each week)? (Hint: The height at the end of week 1 is 3cm, week 2 is 6cm, etc.) | ANSWER: 45 cm
QUESTION: Find the sum of the first 3 terms of a GP where the first term is 10 and the common ratio is 1/2. | ANSWER: 17.5
MCQ
Quick Quiz
Which of these sequences is a Geometric Progression?
2, 4, 6, 8...
3, 6, 12, 24...
1, 2, 3, 4...
5, 10, 15, 20...
The Correct Answer Is:
B
Option B (3, 6, 12, 24...) is a GP because each term is found by multiplying the previous term by 2 (the common ratio). The other options are arithmetic progressions or neither.
Real World Connection
In the Real World
Imagine you are investing money in a Fixed Deposit (FD) at an Indian bank. If the interest is compounded annually, the total amount grows in a Geometric Progression. Calculating the sum of these terms helps you understand how much money you will have after a certain number of years, which is very useful for planning your future savings!
Key Vocabulary
Key Terms
GEOMETRIC PROGRESSION (GP): A sequence of numbers 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 | TERM: Each number in a sequence | SUM: The total value obtained by adding all the terms together
What's Next
What to Learn Next
Great job learning about the sum of n terms! Next, you can explore the 'Formula for the Sum of n terms of a Geometric Progression'. This will teach you a quicker way to find the sum without adding each term individually, especially useful for many terms!
