GP Sum

In a GP sum, every term is obtained by multiplying the previous one by a fixed number called the common ratio. For a better understanding, consider this example where a student deposits ₹100 in the first week, ₹200 in the second week, ₹400 in the third week, and continues this in sequence till 6 weeks. The total amount in six weeks would be ₹6,300. Calculating step by step is easy. To calculate this for 15 weeks, we will be using the formula, 

Sn=a(1-rⁿ) / 1-r  When r is not equals to 1
First term a = ₹100
Common ratio r = 2
Number of terms n = 15

Now, putting the numbers in the formula. 

S15=100(1-2¹⁵) / 1-2 

Calculating 2¹⁵= 32,768, we get:
Substituting the values into the formula:

S_n=100 x 1-2¹⁵ / 1-2

S₁₅=100 x 1-32768 / -1

S₁₅=100 x -32767 / -1

S₁₅=100 x 32767

S₁₅=3,276,700

Therefore, by the end of 15 weeks, the student will have ₹3,276,700.
 

The sum of an infinite GP is the aggregate of its infinitely many terms. This sum is used only if the common ratio r satisfies ∣r∣ < 1, if the terms get smaller and smaller, approaching zero

For calculating the sum of an infinite GP, use this formula:
S=a / 1-r
a is the first term,

r is the common ratio, and

∣r∣<1 for convergence.

Calculate the series:
₹100 + ₹50 + ₹25 + ₹12.5 + ₹6.25 + …
So a = 100, r = 0.5
Now we will put the numbers in the formula:
S=100 / 1-0.5=100 / 0.5=200
As a result, we get the total sum = 200
 

There are two main formulas for the sum of GP

1. Sum of the First n Terms of a Finite GP

For a geometric progression, we define the first term a, the common ratio r, and n terms; the formula for the sum is:
S_n=a(1-rⁿ) / 1-r When r  1
Special Case: If r = 1, the sum can be computed in a simple way:
S_n = a×n

2. Sum of an Infinite GP

For an infinite geometric progression, we define the first term a and the common ratio r, the sum exists only if ∣r∣<1. The sum is:
S=a / 1-r (for ∣r∣<1)

If ∣r∣≥1, the series does not converge and has no finite sum.
 

The sum of the first n terms of a geometric progression (GP) is computed using a formula, depending on the common ratio r. For a GP, we define the first term a, common ratio r, and number of terms n. The formula is:
S_n=a(1-rⁿ) / 1-r When r  1
If r = 1, the sum is:
S_n = a×n

A student is saving money for someone. He saves ₹2 in the first week, ₹4 in the second week, ₹8 in the third week, and goes on till 6 weeks. This doubles his savings each week. 
First term a = ₹2
Common ratio r = 2
Number of terms n = 6
Now we will put the numbers in the formula.

S₆=2(1-2⁶) / 1-2=2(1-64) / -1=2(-63) / -1=126
So after 6 weeks, the student will have a total amount of ₹126.
 

Special cases in geometric progressions (GP) happen when the common ratio (r) takes specific values; this modifies the series' behavior and convergence.

Special Cases in Geometric Progression

Finite GP with r = 1:

When the common ratio r is equal to 1, each term will be the same. The sum of the first n terms formula is:
S_n = an Where we consider a as the first term.

Infinite GP with ∣r∣<1:

For an infinite geometric progression to converge, the absolute value of the common ratio must be less than 1. The sum of an infinite GP is:
S=a /1-r
This formula will be valid only when ∣r∣<1; otherwise, the sequence will change.
 

 Geometric progressions (GPs) have many real-life applications where numbers change at a constant rate, making them important in fields like finance, biology, and technology.

• Ball Bouncing: When you drop a ball, it bounces back to a certain fraction of its previous height. If it bounces back to half its height each time, the heights of the bounces follow a geometric progression: 100%, 50%, 25%, 12.5%, ... The total distance the ball travels is the sum of these heights.

• Saving Pocket Money: Imagine you save ₹20 in your piggy bank this week. Next week, you plan to save double that amount, ₹40. Your savings follow a pattern: 20, 40, 80, ... This is a geometric progression where each amount is double the previous one. To know how much you have saved after a certain number of weeks, you can use the sum of a geometric progression.

• Puzzle Pieces: In some puzzles, the number of pieces doubles each time you add a new section. Suppose you start with 1 piece, then 2, then 4, then 8, and so on. This doubling follows a geometric progression, and knowing this helps you understand how big the puzzle will get.

• Tree Branching: Trees branch out in a way that each branch splits into multiple smaller branches. If each branch splits into two, the total number of branches at each level follows a geometric progression: 1, 2, 4, 8, ... This pattern helps scientists understand how trees grow and spread.

• Game Points: In some video games, you earn points that double each time you complete a task. For example, earning 10 points, then 20, then 40, and so on. This doubling follows a geometric progression. Knowing this pattern helps you understand how quickly your score can increase.