The sum of first three terms of a G.P. is 1312 and their product is – 1.Find the common ratio and the terms

Sure, let's solve the problem step by step.

Step 1: Let's assume that the first term of the geometric progression (G.P.) is "a" and the common ratio is "r".

Step 2: The sum of the first three terms of the G.P. is given as 1312. Using the formula for the sum of a G.P., we can write the equation as: a + ar + ar^2 = 1312

Step 3: The product of the first three terms of the G.P. is given as -1. Using the formula for the product of a G.P., we can write the equation as: a * ar * ar^2 = -1

Step 4: Simplify the equations: a(1 + r + r^2) = 1312 a^3r^3 = -1

Step 5: Divide the second equation by the first equation to eliminate "a": (a^3r^3) / (a(1 + r + r^2)) = -1 / 1312

Step 6: Simplify the equation further: r^2 - r + 1 = -1 / 1312

Step 7: Multiply both sides of the equation by 1312 to get rid of the fraction: 1312r^2 - 1312r + 1312 = -1

Step 8: Rearrange the equation: 1312r^2 - 1312r + 1313 = 0

Step 9: Solve the quadratic equation using the quadratic formula: r = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1312, b = -1312, and c = 1313.

Step 10: Substitute the values into the quadratic formula and solve for "r": r = (-(-1312) ± √((-1312)^2 - 4 * 1312 * 1313)) / (2 * 1312)

Simplifying the equation further will give us the values of "r".

Step 11: Once we have the value of "r", we can substitute it back into the equation in Step 2 to find the value of "a".

Step 12: Finally, we can find the three terms of the G.P. by using the values of "a" and "r" in the equation a, ar, and ar^2.

I hope this helps you solve the problem!