given a Gp below, find the values of m, and the next 2 terms:
m, m+3, m^2+3
To find the values of m and the next 2 terms in the geometric progression (GP) m, m+3, m^2+3, we can utilize the properties of a geometric progression.
A geometric progression is a sequence in which each term after the first is found by multiplying the previous term by a constant called the common ratio (r). In our case, we need to determine the value of m and find the common ratio.
From the given GP sequence, we have the following terms:
Term 1: m
Term 2: m + 3
Term 3: m^2 + 3
To find the common ratio, we can divide Term 2 (m + 3) by Term 1 (m):
(m + 3) / m
For a geometric progression, this ratio should be the same for all consecutive terms. Therefore, we also need to check if the ratio of Term 3 (m^2 + 3) to Term 2 (m + 3) is the same:
(m^2 + 3) / (m + 3)
Since these two ratios should be equal, we can equate them:
(m + 3) / m = (m^2 + 3) / (m + 3)
To solve this equation for m, we can cross-multiply and simplify:
(m + 3) * (m + 3) = m * (m^2 + 3)
(m^2 + 6m + 9) = m^3 + 3m
m^3 + 3m - m^2 - 6m - 9 = 0
m^3 - m^2 - 3m - 9 = 0
We now have a cubic equation. To find the values of m, you can use various methods such as factoring, synthetic division, or numerical methods like Newton-Raphson.
Once you have found the values of m, substitute them back into the original GP sequence to find the next two terms.