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The first term of an Arithmetic Progression (AP) is equal to the first term of a Geometric Progression (GP). The second term of the AP is equal to the fourth term of the GP, while the tenth term of the AP is equal to the seventh term of the GP.

(a) Given that \( a \) is the first term and \( d \) is the common difference of the AP while \( r \) is the common ratio of the GP, write two equations connecting the AP and the GP.

(b) Find the value of \( r \) that satisfies the progressions.

(c) Given that the tenth term of the GP is 5120, find the values of \( a \) and \( d \).

(d) Calculate the sum of the first 20 terms of the AP.

Answer :

The problem involves deriving relations between terms of an Arithmetic and Geometric Progression, followed by determining the common ratio of the GP, and finally calculating specific terms and the sum of a sequence.

The student is presented with a problem regarding an Arithmetic Progression (AP) and a Geometric Progression (GP). We are given certain conditions that relate terms from both progressions. Let's address each part of the question.

(a) Equations connecting AP and GP

For AP, we have the n-th term given by: a_n = a + (n-1)d, where a is the first term and d is the common difference. For GP, the n-th term is given by: g_n = ar^{n-1}, where r is the common ratio. Given the problem's conditions, we arrive at two equations:

(b) Value of r

Subtracting the first equation from the second, we get 8d = ar^6 - ar^3 or 8d = ar^3(r^3 - 1). Ratio r thus satisfies the equation r^3 - 1 = 8d/ar^3. From the first equation, we have r^3 = 1 + d/a. Replacing in the second equation, we get (1+d/a) - 1 = 8d/a(1+d/a), leading to r = 2.

(c) Values of a and d

Knowing the 10th term of GP (5120) and r (2), we can write ar^9 = 5120, which gives us a. We also have the value of r to use in a + d = ar^3 to find d.

(d) Sum of the first 20 terms of the AP

The sum of the first 'n' terms of an AP is given by: S_n = n/2(2a + (n-1)d). Substituting the known values of a and d, we calculate S_20 for the sum of the first 20 terms.

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