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A geometric progression (G.P.) is a sequence where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number known as the common ratio. For example, in the sequence 2, 4, 8, 16..., the common ratio is 2, since each term is the previous term multiplied by 2.
To express this mathematically, if the first term of the G.P. is denoted by \(a_1\) and the common ratio by \(r\), then the \(n^{th}\) term can be represented as:

• Second term: \(a_2 = a_1 \times r = a_1 \times r\)
• Third term: \(a_3 = a_2 \times r = a_1 \times r^2\)
• \(n^{th}\) term: \(a_n = a_1 \times r^{(n-1)}\)

Understanding G.P. is crucial as it shows how exponential growth can occur, such as in the case of compound interest or population growth. In the problem, the terms \(a, b, c\) form a G.P., meaning their ratios are constant, emphasizing the predictability and regularity of a G.P.

The common difference is a key feature of arithmetic progression (A.P.). It is the difference between any two successive terms of an A.P., and it remains constant throughout the sequence. For example, in the A.P. 3, 6, 9, 12..., the common difference is 3, as each term differs from the previous one by this constant amount.
If we denote the first term by \(a_1\) and the common difference by \(d\), the formula for the \(n^{th}\) term in an A.P. is given by:

• \(n^{th}\) term: \(a_n = a_1 + (n-1)d\)

In the exercise, terms \(a, b, c\) are the 7th, 11th, and 13th terms of an A.P., respectively. This consistent difference forms the "backbone" of any arithmetic sequence, ensuring the increments between terms do not change.
Recognizing this consistent pattern allows us to establish relationships between terms and solve problems involving finding unknown values within the progression.

The concept of a common ratio is fundamental to understanding a geometric progression (G.P.). In any G.P., the common ratio \(r\) is the factor by which we multiply a term to get the next term in the sequence. For instance, in the sequence 5, 10, 20, 40..., the common ratio is 2. Each term is 2 times the preceding one.
In this sequence:

• First term \(a_1\), second term: \(a_2 = a_1 \times r\)
• Third term: \(a_3 = a_2 \times r = a_1 \times r^2\)
• Similarly, \(n^{th}\) term: \(a_n = a_1 \times r^{(n-1)}\)

In the given problem, the terms \(a, b, c\) must satisfy the condition of a G.P. This means \(\frac{b}{a} = \frac{c}{b}\), illustrating how the common ratio governs the relationship between any three consecutive terms.
This fixed multiplicative rule enables us to solve and predict values at various positions within a G.P., adding an exponential twist to sequences compared to arithmetic progressions, which are additive in nature.