In mathematics, geometric progression is a special series, and the ratio of each item to the previous item remains unchanged, which is called common ratio. As for the sequence TN = √ n (n (the root of n is the power of n), let's discuss whether it satisfies the properties of geometric series.
First of all, we will write the first few items of the sequence TN = √ n (n (the root of n is the power of n):
t 1 = √ 1 = 1
t2 = √2
t3 = √3
t4 = √4 = 2
t5 = √5
t6 = √6
...
If we calculate the ratio between two adjacent terms, we can get:
t2 / t 1 = √2 / 1 = √2
t3 / t2 = √3 / √2
t4 / t3 = 2 / √3
t5 / t4 = √5 / 2
t6 / t5 = √6 / √5
...
As can be seen from the above calculation, the ratio between two adjacent terms is not constant, that is, the sequence tn = √n does not satisfy the property of geometric progression. So tn = √n is not a geometric series.
Geometric series requires the ratio of two adjacent terms to be constant, but for the sequence tn = √n, the ratio of two adjacent terms will change with the increase of n, so it does not conform to the definition of geometric series. This also reflects the nature and characteristics of different series in mathematics, and we need to pay attention to the differences between different series when learning series.