The sum of first three terms of a G.P. is 1213 and their product is -1 . Find the common ratio and the terms.
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Step-by-Step Solution
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 a3=−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 1213. 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 12r2+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.