Geometric Sequences (Hard)

Late AMC 10/12 through early AIME — dense computation.

Geometric Sequences: core idea

Infinite

|r|<1

A geometric sequence has $a_2=6,a_5=162$ . Find positive $r$ .

Hint: Use ratios for terms and the finite or infinite geometric-sum formula. Answer: 3.

1. Find $1+3+9+\cdots+3^8$ .
2. A ball rebounds to $3/4$ of its height. Dropped from $64$ , find fourth-bounce height.
3. Three consecutive geometric terms have product $1728$ . Find the middle term.
4. Find the first ten-term sum of $5,10,20,\ldots$ .
5. $a_1=81,a_5=1$ . Find positive $r$ .
6. Find $8+4+2+\cdots$ .
7. $a_3=20,a_6=160$ . Find $a_1$ .
8. $x,12,27$ are consecutive positive geometric terms. Find $x$ .
9. A culture starts at $3$ and doubles hourly. How many after $8$ hours?