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A geometric progression, or G.P., 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. If you have a sequence like this, it will look something like: \

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• First Term: \( a \)
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• Second Term: \( ar \)
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• Third Term: \( ar^2 \)
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• Fourth Term: \( ar^3 \)
\

\Where \( a \) is the initial term and \( r \) is the common ratio. Each subsequent term is obtained by multiplying the previous term by the common ratio \( r \).
A geometric progression exemplifies exponential growth or decay, depending on whether \( r \) is greater than or less than 1. If \( r \) is greater than 1, the sequence grows, while it diminishes if \( r \) is between 0 and 1.
In the given exercise, the terms of an arithmetic progression (A.P.) are organized into a G.P., which means their ratios maintain this fixed proportion.

The common difference is a central concept when dealing with arithmetic progressions (A.P.). In an A.P., each term is the result of adding a constant value, called the common difference, to the previous term.
The general formula for the \(n^{th}\) term of an A.P. is: \( a_n = a + (n-1)d \), where \( a \) is the first term, and \( d \) is the common difference. This results in a sequence like:

• First Term: \( a \)
• Second Term: \( a + d \)
• Third Term: \( a + 2d \)
• Fourth Term: \( a + 3d \)

The characteristic feature of an arithmetic progression is that this difference between successive terms is always the same.
In solving the exercise, this concept helps us structure and manipulate terms to find relationships, such as showing that \( p-q, q-r, r-s \) form their own mathematical pattern.

Sequences and series constitute fundamental concepts in mathematics. A sequence is an ordered list of numbers following a specific pattern, while a series is the sum of terms in a sequence.

• Arithmetic Sequence (A.P.): A sequence with a constant difference between successive terms, such as 2, 4, 6, where the difference is 2.
• Geometric Sequence (G.P.): A sequence with a constant factor between successive terms, such as 3, 6, 12, where the ratio is 2.
• Series: The sum of the terms of a sequence. In an A.P., this is called an arithmetic series, whereas in a G.P., it’s a geometric series.

Understanding these concepts involves recognizing patterns and utilizing formulas to find specific terms or sums. For instance, when the exercise states that terms of an A.P. form a G.P., it implies a special condition aligning the terms through multiplication.
By solving the problem, we find how these kinds of sequences interact, culminating in a relationship where \( p-q, q-r, r-s \) form their arithmetic progression.