Geometric Sequences (Medium)

AMC 10/12 mid (≈Q9–16) — multi-step.

Geometric Sequences: core idea

Finite sums.

Find the sum of the first $6$ terms of $1,2,4,\ldots$ .

Hint: Use the finite or infinite geometric-sum formula. Answer: 63.

1. In $3,6,12,\ldots$ , which term is $384$ ?
2. A geometric sequence has $a_2=6$ and $a_5=162$ . Find $r$ .
3. Find the infinite sum $1+\frac12+\frac14+\cdots$ .
4. Find the sum $5+15+45+135$ .
5. A ball bounces to half its previous height, beginning at $16$ . Find its height after $4$ bounces.
6. Find $a_1$ if $a_3=36$ and $r=6$ .
7. Find the product of the first $4$ terms of $2,4,8,16,\ldots$ .
8. Find the infinite sum $6+3+\frac32+\cdots$ .
9. A sequence has $a_1=2$ and $a_4=54$ . Find $a_2$ if the ratio is positive.