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In an infinite GP, every term is 7 times the sum of all terms that follow. If sum of first and second terms is 9, find the first term of G.P.

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In an infinite GP, every term is 7 times the sum of all terms that fol...

Given Information:
- Sum of first and second terms = 9
- Each term is 7 times the sum of all terms that follow

Let's assume the first term of the GP is 'a' and the common ratio is 'r'. Therefore, the second term will be ar.

Sum of first and second terms:
a + ar = 9
a(1 + r) = 9

Each term is 7 times the sum of all terms that follow:
a = 7(ar + ar^2 + ar^3 + ...)
a = 7ar(1 + r + r^2 + ...)
a = 7ar/(1 - r)

Substitute the value of a from the second equation into the first equation:
7ar/(1 - r) = 9
7r = 9 - 9r
16r = 9
r = 9/16

Substitute the value of r back into the equation a = 7ar/(1 - r):
a = 7*(9/16)/(1 - 9/16)
a = 63/16*(16/7)
a = 9

Therefore, the first term of the GP is 9.
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Given sequence: 3, 11, 34, 69, 131, 223To find the pattern in the sequence, let's calculate the differences between consecutive terms:11 - 3 = 834 - 11 = 2369 - 34 = 35131 - 69 = 62223 - 131 = 92The differences between consecutive terms are not following a consistent pattern. Therefore, the given sequence does not have a simple arithmetic or geometric progression.However, if we observe the differences between the differences, we can see a pattern:23 - 8 = 1535 - 23 = 1262 - 35 = 2792 - 62 = 30The differences between the differences are increasing by a constant value. This indicates that the given sequence might have a polynomial pattern.To find the polynomial pattern, we can use the method of finite differences. Let's calculate the second differences:15 - 12 = 312 - 27 = -1527 - 30 = -3The second differences are not constant. Therefore, the given sequence might have a polynomial of degree 2.Let's assume the polynomial is of the form ax^2 + bx + c. Substituting the given terms into the polynomial, we can form the following equations:1. 3a + b + c = 32. 9a + 3b + c = 113. 16a + 4b + c = 34Solving these equations, we get a = 1, b = 2, and c = 0.Therefore, the polynomial pattern for the given sequence is x^2 + 2x.Now, let's find the missing terms in the sequence

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In an infinite GP, every term is 7 times the sum of all terms that follow. If sum of first and second terms is 9, find the first term of G.P.?
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