7.3 Geometric sequences

Which of the following expressions is/are the general form of the geometric sequences [math]a_n[/math] that satisfies [math]a_3=4[/math] and [math]r=-3[/math]?

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]

(Mid-term) Find the constant [math]k[/math] such that [math]2k+5, k+3, 1[/math] form a geometric sequence.

(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].

(Calculator) (Loan problem) Suppose you borrow [math]1,000,000[/math] dollars from the bank to take Mr. Fang’s class (very touching). If the interest rate is [math]1\%[/math], and you want to pay back in [math]10[/math] years evenly. Let [math]x[/math] be your payment each year. Observe the pattern: ([math]P=1,000,000, R=1.01[/math])
[math]\begin{array}{ll}
1\text{ year}&PR-x=0\\&\\
2\text{ years}& (PR-x)R-x=0\quad\Rightarrow\quad PR^2-(1+R)x=0\\&\\
3\text{ years}& ((PR-x)R-x)R-x=0\quad\Rightarrow\quad PR^3-(1+R+R^2)x=0\\&\\
\vdots& \vdots\\&\\
\end{array}[/math]
Use this pattern to find that how much you pay back to the bank each year.

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].

Let [math]\{a_n\}[/math] be a geometric sequence with common ratio [math]r[/math]. If [math]\{b_n\}[/math] is a new sequence that satisfies
[math]\bullet[/math] [math]b_1=a_1+a_2+a_3[/math]
[math]\bullet[/math] [math] b_2=a_4+a_5+a_6[/math]
[math]\bullet[/math] [math] b_3=a_7+a_8+a_9[/math]
[math]\bullet[/math] [math] \vdots[/math]
Which of the following statements is true?

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?

Consider a polynomial equation [math]x^3-3x^2-ax+8=0[/math]. Suppose [math]p, q, r[/math] are the solutions of the equation and [math]p, q, r[/math] form a geometric sequence. Given that [math]p<r[/math], find the constants [math]p, q, r[/math] and [math]a[/math]. Hint: Expand [math](x-p)(x-q)(x-r)[/math] and compare the coefficients.

Consider a geometric sequence [math]\{a_n\}[/math] that satisfies [math]a_1+a_2+a_3=18[/math] and [math]a_2+a_3+a_4=-9[/math]. Find the infinite series of [math]\{a_n\}[/math].