Geometric sequences

Summary

The general term of a geometric sequence is given by $T_{n}=ar^{n-1}$. Here:

Class exercises

  1. Consider a degenerate case in which the common ratio of a geometric sequence is $1$. Describe the general appearance of the geometric sequence in this case.
  2. Explain why zero is not allowed to be one of the terms in a geometric sequence.
  3. Identify the first term and the common ratio of each geometric sequence:
    • $-2,4,-8,16,-32,\cdots$
    • $\frac{-1}{4},\frac{1}{12},\frac{-1}{36},\frac{1}{108},\frac{-1}{324},\cdots$
    • $0.8,2.4,7.2,21.6,64.8,\cdots$
  4. The first term of a geometric sequence is $7$, while the common ratio is $5$. Find the $10$th term of the sequence.
  5. Find the number of terms in the sequence $2,-6,18,-54,\cdots,39366$.
  6. Insert five geometric means between $3$ and $192$. That is, find numbers $a,b,c,d,e$ for which the sequence $3,a,b,c,d,e,192$ would be geometric.
  7. Let $x\neq -1,0$ be a real number. PROVE that the 3-term sequence $$1,~(x+1),~(x^2+2x+1)$$ is geometric.
  8. Let $x,y$ be non-zero real numbers. PROVE that the enumeration $$2x,~x+y,~y$$ cannot be geometric.
  9. Let $a,b,c,d\neq 0$. PROVE that if the sequence $a,~b,~c,~d$ is geometric, then $(b+c)^2=b(c+d)+c(a+b)$.
  10. Let $a,b,c\neq 0$. PROVE that the quadratic equation $ax^2+bx+c=0$ has only one root if, and only if, the sequence $2a,~b,~2c$ is geometric.