So [itex]s_2> a s_n[/itex], [itex]s_3> a s_2> a(a s_1)= a^2 s_1[/itex], [itex]s_4> a s_3> a(a^2 s_1)= a^3 s_1[/itex]. So [itex]s_n> [/itex] a to what power times [itex]s_1[/itex]? What does that have to do with a "geometric sequence"?StarTiger said:Homework Statement
Suppose that a sequence {s_n} of positive numbers satisfies the condition s_(n+1) > αs_n for all n where α > 1. Show that s_n → ∞
My teacher mentioned something about making it into a geometric sequence and taking the log. I'm just confused.
Homework Equations
The Attempt at a Solution
When a sequence satisfies a condition for all n, it means that the condition is true for every term in the sequence, no matter how large or small n (the term number) is.
One example is the sequence 2^n, where n is a term number. This sequence satisfies the condition that each term is equal to 2 raised to the power of the term number, for all n.
To prove that a sequence satisfies a condition for all n, you can use mathematical induction. This involves showing that the condition holds for the first term in the sequence, and then assuming it holds for some arbitrary term k, and using that assumption to show that the condition also holds for the next term k+1. This process can be repeated indefinitely, proving that the condition holds for all terms in the sequence.
Sequences satisfying a condition for all n are important in mathematics because they allow us to make generalizations and prove properties about the sequence as a whole. By knowing that a condition holds for all terms in the sequence, we can confidently make statements about the behavior and properties of the sequence, which can then be applied to solve problems and make predictions in various fields of mathematics.
Yes, a sequence can satisfy multiple conditions for all n. For example, the sequence n^2 satisfies the conditions that each term is a perfect square and that each term is greater than the previous term, for all n. In general, a sequence can satisfy any number of conditions as long as they hold for all terms in the sequence.