The first term of an infinite geometric progression (GP) is equal to 1 according to the provided reference. This is a specific scenario detailed in the reference text where the first term is explicitly stated to be 1, and each subsequent term is the sum of all succeeding terms.
Understanding Infinite Geometric Progressions
Before delving into the specifics of this particular GP, let's briefly review the concept of an infinite GP:
- An infinite geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (often denoted as 'r').
- The general form of a GP is: a, ar, ar2, ar3, ... where 'a' is the first term.
- An infinite GP can either converge to a sum (if |r| < 1) or diverge (if |r| >= 1).
Analysis of the Given Scenario
The reference provides a unique case:
The first term of an infinite G.P. is 1 and any term is equal to the sum of all the succeeding terms.
This implies a very specific relationship between the terms. Let's represent the terms of this particular GP as:
- a1 = 1 (the first term)
- a2 = the sum of all terms from a3 onwards
- a3 = the sum of all terms from a4 onwards, and so on.
Deriving the Common Ratio
If we denote the common ratio as 'r', we can write the GP as: 1, r, r2, r3,...
From the provided information, we know that:
- The first term is 1.
- The second term, 'r', is equal to the sum of all succeeding terms: r = r2 + r3 + r4 + ....
This is a special case of the sum of an infinite GP, which can be written as:
r = r2 / (1-r), (assuming |r| < 1)
r (1 - r) = r2
r - r2 = r2
2r2 - r = 0
r (2r - 1) = 0
The possible values for r are 0 and 1/2. r cannot be 0 since it must be a geometric progression, and r must be less than 1 for this to be a converging infinite GP, therefore, r = 1/2.
So the GP is 1, 1/2, 1/4, 1/8, ...
Verifying the Condition
Let’s check if the condition that any term equals the sum of all subsequent terms holds true for our sequence:
- The second term (1/2) equals the sum of 1/4 + 1/8 + 1/16 + ... = (1/4) / (1-1/2) = 1/2
- The third term (1/4) equals the sum of 1/8 + 1/16 + 1/32 + ... = (1/8) / (1-1/2) = 1/4
This verifies the relationship, confirming that our common ratio and series fit the unique condition described.
Conclusion
In the specific infinite geometric progression described in the reference, where each term is the sum of all succeeding terms, the first term is equal to 1. The common ratio of this specific GP is 1/2, resulting in the series: 1, 1/2, 1/4, 1/8, and so on.
