Final answer:
To find G1 and r in a geometric progression (GP), we can use the given equations and solve for the unknowns. G1 = 48 / (r^2 - 1) and r = 48 / (G1 * r^4).
Step-by-step explanation:
To find the value of G1 and r in a geometric progression (GP) with given conditions, we can use the formula for the nth term of a GP: Gn = G1 * r^(n-1). We are given two equations: G7 - G5 = 48 and G6 + G5 = 48. Substituting the values, we get two equations: G1 * r^6 - G1 * r^4 = 48 and G1 * r^5 + G1 * r^4 = 48. Simplifying these equations, we can solve for G1 and r.
From the first equation, we can factor out G1 * r^4: G1 * r^4 * (r^2 - 1) = 48. Since G1 * r^4 cannot be zero, we can divide both sides by (r^2 - 1) to get G1 = 48 / (r^2 - 1). And from the second equation, we can factor out G1 * r^4: G1 * r^4 * (r + 1) = 48. Again, assuming G1 * r^4 is not zero, we can divide both sides by (r + 1) to get r = 48 / (G1 * r^4).