The sum of first three terms of a G.P. is $\frac{13}{12}$ and their product is -1 . Find the common ratio and the terms.

Step 1: Define terms and use product information

Let the first three terms of the Geometric Progression (G.P.) be $a/r$, $a$, and $ar$, where $a$ is the middle term and $r$ is the common ratio. The problem states that their product is -1. We can use this information to find the value of $a$.

Step 2: Solve for 'a'

Multiplying the three terms, we get $a^3 = -1$. Taking the cube root of both sides, we find that $a = -1$. This is the middle term of the G.P.

Step 3: Use sum information

The problem also states that the sum of the first three terms is $\frac{13}{12}$. We can substitute the value of $a = -1$ into this equation.

Step 4: Substitute 'a' and simplify

Substitute $a = -1$ into the sum equation. Then, multiply the entire equation by $12r$ to eliminate the denominators and rearrange it into a quadratic equation.

Step 5: Solve the quadratic equation for 'r'

The quadratic equation $12r^2 + 25r + 12 = 0$ can be factored as $(4r+3)(3r+4) = 0$. This gives two possible values for the common ratio, $r = -3/4$ or $r = -4/3$.

Step 6: Find the terms for each value of 'r'

For each value of $r$, we substitute it back into the expressions for the three terms ($a/r$, $a$, $ar$) with $a = -1$. This yields two possible sets of terms for the G.P.

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