Sum of Possible Values of First Term of GP Linked to AP

To find the sum of all possible values of the first term of the Geometric Progression (G.P.), let us define the given variables:

Let:

• $a$ be the first term of the A.P.
• $d$ be the common difference of the A.P.
• $b$ be the first term of the G.P.
• $r$ be the common ratio of the G.P.

From the problem statement, we are given the following relationships between the parameters:

1. The first term of the A.P. is equal to the common ratio of the G.P.: $a = r$
2. The first term of the G.P. is equal to the common difference of the A.P.: $b = d$

We are also given the conditions for the sums of the terms:

Condition 1: The sum of the first ten terms of the A.P. is $160$. $S_{10} = \frac{10}{2} [2a + 9d] = 160$ $5(2a + 9d) = 160$ $2a + 9d = 32$

Substituting $a = r$: $2r + 9d = 32 \quad \text{--- (1)}$

Condition 2: The sum of the first two terms of the G.P. is $8$. $b + br = 8 \implies b(1 + r) = 8$

Substituting $b = d$: $d(1 + r) = 8 \quad \text{--- (2)}$

---

Now, we solve for $d$ (which is equal to the first term of the G.P., $b$):

From equation (1), express $r$ in terms of $d$: $r = \frac{32 - 9d}{2}$

Substitute this expression for $r$ into equation (2): $d \left( 1 + \frac{32 - 9d}{2} \right) = 8$ $d \left( \frac{2 + 32 - 9d}{2} \right) = 8$ $d(34 - 9d) = 16$ $34d - 9d^2 = 16$ $9d^2 - 34d + 16 = 0$

Check the discriminant ($D$) of this quadratic equation to confirm real roots: $D = (-34)^2 - 4(9)(16) = 1156 - 576 = 580 > 0$

Since $D > 0$, the quadratic equation $9d^2 - 34d + 16 = 0$ has two distinct real roots, $d_1$ and $d_2$.

By Vieta's formulas, the sum of the roots of a quadratic equation $Ax^2 + Bx + C = 0$ is given by $-\frac{B}{A}$: $\text{Sum of possible values of } d = d_1 + d_2 = -\frac{-34}{9} = \frac{34}{9}$

Since $b = d$, the sum of all possible values of the first term of the G.P. is: $\frac{34}{9}$

Correct Answer: Option A ($\frac{34}{9}$)