7.3 Geometric sequences (in class)

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]?

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

(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. (integer)

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?

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.