7.3 Geometric sequences practice 1

Which of the following expressions is/are the general form of the geometric sequences [math]a_n[/math] that satisfies [math]a_4=2[/math] and [math]a_2=4[/math]?

(Calculator) Find the series of the following geometric sequences.

(a) [math]a_n=2022\cdot2^{-n}[/math]. [math]S_{10}[/math]

(b) [math]a_1=3[/math] and [math]\{a_n\}[/math], and this sequence is also arithmetic. [math]S_{2022}[/math]

(c) [math]a_2=-5, r=-2[/math]. [math]S_7[/math]

(Calculator) Consider a polynomial function [math]f(x)=1+x^2+x^4+x^6+\cdots+x^{10}[/math]. Find [math]f(\sqrt{3})[/math].

Let [math]\{a_n\}[/math] be a geometric sequence. If [math]a_1=100[/math] and [math]a_2+a_3=12a_1[/math], find [math]a_4[/math]. Write your answer in ascending order.

Let [math]\{a, b, c\}[/math] be a geometric sequence. If [math]abc=125[/math], what is the value of [math]b[/math]?

(Calculator) Let [math]\{a_n\}[/math] be a sequence satisfying [math]a_n=\dfrac{3^n}{10000}[/math].

(a) Find the minimum of [math]m[/math] such that [math]a_m\geq1[/math].

(b) Find the minimum of [math]n[/math] such that [math]S_n\geq1[/math] where [math]S_n=a_1+a_2+\cdots+a_n[/math].

If [math]a_n[/math] is a geometric sequence that satisfies [math]a_2=18[/math] and [math]a_4=9[/math], which of the following statments is/are true?

Let [math]a_n[/math] be a geometric sequence but not an arithmetic sequence. If [math]a_1=a_3=7[/math], which of the following statments is/are true?

Consider a set [math]S=\{1, 2, 3, \cdots, 9\}[/math]. Choose 3 numbers (the same number can be chosen more than once) to form a sequence, how many different geometric sequences can be made?

Let [math]\{a_n\}[/math] be a geometric sequence and [math]k[/math] be a nonzero constant. Which of the following sequences are also geometric?