Understanding Geometric Progression (G.P.)
In a geometric progression, the ratio between consecutive terms remains constant. Given three terms, 5, 9, and 11, we need to determine how many G.P.s can be formed using these numbers.
Forming the G.P.
1. Definition of G.P.:
A sequence of numbers is in G.P. if there exists a common ratio 'r' such that:
- Second term = First term * r
- Third term = Second term * r
2. Finding Ratios:
Let's denote the first term as 'a'. Then we can express the terms:
- Second term: 5 * r = 9 (from the first term)
- Third term: 9 * r = 11 (from the second term)
3. Setting Up Equations:
From the above equations, we can derive:
- r = 9/5
- r = 11/9
4. Equating Ratios:
For these to be valid, both expressions for r must equal:
- 9/5 = 11/9
5. Cross Multiplying:
If we cross-multiply:
- 9 * 9 = 5 * 11
- 81 = 55 (which is false)
Conclusion
Since we have established that the ratios do not equate, it is impossible to form a G.P. with the terms 5, 9, and 11.
Final Answer
Thus, the number of G.P.s having 5, 9, and 11 as its three terms is zero.