The sum of of first ten terms of an A.P. is equal to 155 and...

The sum of of first ten terms of an A.P. is equal to 155 and the sum of first two terms of a G.P. is 9 . If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to the common difference of the A.P, then (A) first term of the G.P. is $\frac{2}{3}, 3$ (B) first term of the A.P. is $\frac{2}{3}, 3$ (C) Common ratio of the G.P. is $\frac{25}{2}, 2$ (D) Common difference of the A.P is $\frac{2}{3}, 3$

The sum of of first ten terms of an A.P. is equal to 155 and the sum of first two terms of a G.P. is 9 . If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to the common difference of the A.P, then
(A) first term of the G.P. is $\frac{2}{3}, 3$
(B) first term of the A.P. is $\frac{2}{3}, 3$
(C) Common ratio of the G.P. is $\frac{25}{2}, 2$
(D) Common difference of the A.P is $\frac{2}{3}, 3$

Key Concepts

Arithmetic Progression (AP)

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference, known as the common difference, defines the progression and is used in formulas to determine any term in the sequence as well as the sum of a certain number of terms. The sum of the first n terms of an AP can be calculated using the formula S = n/2 × (2a + (n-1)d), where a is the first term and d is the common difference.

Geometric Progression (GP)

A geometric progression is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. For a GP, the sum of the first n terms depends on whether the common ratio is equal to 1 or not, and is given by the formula S = a(1?r^n)/(1?r) when r ? 1, where a is the first term and r is the common ratio. For just the first two terms, the sum is simply a + ar.

Relationship between AP and GP Parameters

In problems where parameters of an arithmetic progression and a geometric progression are intertwined, specific conditions are given that relate the first term and common difference of the AP with the first term and common ratio of the GP. This requires setting up equations where one sequence’s term plays a role in the other sequence’s definition. Recognizing these relationships is crucial in forming the system of equations needed to solve the problem.

Using Sum Formulas to Form Equations

The key to solving such hybrid problems lies in correctly applying the sum formulas for both arithmetic and geometric progressions. By substituting the relationships between the sequences’ parameters into the sum formulas, one can derive equations that allow the determination of unknown values. This process involves careful algebraic manipulation and an understanding of how the formulas operate.

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