Arithmetic 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).

Arithmetic Sequences: core idea

Competition twists.

**(Proof track.)** Give a complete justification. An arithmetic sequence has $a_4=17,a_{19}=62$ . Find $S_{22}$ .

Hint: Derive the needed term or sum from the defining constant difference. Answer: 869/2.

No proof submission required — answers only.

1. **(Proof track.)** Give a complete justification. If $S_n=n(3n+1)/2$ , find $a_{20}$ .
2. **(Proof track.)** Give a complete justification. Three arithmetic terms sum to $39$ and product to $2145$ . Find the middle term.
3. **(Proof track.)** Give a complete justification. Find the sum of multiples of $9$ from $100$ to $999$ .
4. **(Proof track.)** Give a complete justification. If $a_2a_8=192$ and $a_5=14$ , find $d^2$ .
5. **(Proof track.)** Give a complete justification. An arithmetic sequence has $S_{10}=100,S_{20}=300$ . Find $a_1$ .
6. **(Proof track.)** Give a complete justification. Find the number of terms in $17+23+\cdots+1001$ .
7. **(Proof track.)** Give a complete justification. If $a_1=7$ and $S_n=777$ , with $d=2$ , find $n$ .
8. **(Proof track.)** Give a complete justification. Find the sum of all integers from $1$ to $500$ not divisible by $5$ .
9. **(Proof track.)** Give a complete justification. An arithmetic sequence has $a_1+a_2+a_3=42$ . Find $a_2$ .