Geometric Sequences (AIME / USAJMO / AMO)

Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.

Proof track: treat each problem as a full olympiad writeup on paper. Here you only enter the final answer (integer or short expression).

Geometric Sequences: core idea

Product tricks.

**(Proof track.)** Give a complete justification. A geometric sequence has $a_1=2$ and $S_6=126$ . Find positive ratio.

Hint: Use multiplication of ratios and distinguish a finite sum from a limit. Answer: 2.

No proof submission required — answers only.

1. **(Proof track.)** Give a complete justification. Find $\sum_{k=0}^7 2^k3^{7-k}$ .
2. **(Proof track.)** Give a complete justification. If $a_2=12,a_4=48$ , find $a_6$ .
3. **(Proof track.)** Give a complete justification. A series starts $1/2+1/4+\cdots$ . Find the least $n$ with $S_n>0.99$ .
4. **(Proof track.)** Give a complete justification. Three positive geometric terms sum to $14$ and product $8$ . Find middle term.
5. **(Proof track.)** Give a complete justification. A population triples each year; it is $1458$ after six years. Find initial population.
6. **(Proof track.)** Give a complete justification. Find the product of the first five terms of $3,6,12,\ldots$ .
7. **(Proof track.)** Give a complete justification. If $a_1=81,r=1/3$ , find the sum of all terms.
8. **(Proof track.)** Give a complete justification. Find the number of positive integer geometric sequences with first term $1$ , fourth term $64$ .
9. **(Proof track.)** Give a complete justification. A geometric sequence has $a_2=6,a_4=54$ . Find $a_1$ .