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Arithmetic Sequences (Hard)

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

Easy Medium Hard AIME / USAJMO / AMO

Arithmetic Sequences: core idea

Nested conditions.

Example problem: hint and answer

An arithmetic sequence has $a_3=11,a_{12}=38$ . Find $a_{20}$ .

Hint: Write the term or sum formula before substituting. Answer: 62.

Practice (9 problems — type your answers)

1. $S_{30}=765$ and $a_1=4$ for an arithmetic sequence. Find $d$ .
2. If $a_5+a_{15}=50$ , find $a_{10}$ .
3. How many positive multiples of $7$ are below $1000$ ?
4. The arithmetic mean of $8,x,26$ is $17$ . Find $x$ .
5. Sequence $3,8,13,\ldots$ : find least $n$ with $a_n>500$ .
6. How many terms are in $4+7+10+\cdots+100$ ?
7. For $a_1=2,d=2$ , find $n$ if $S_n=210$ .
8. $a_7=2a_2$ and $a_1=3$ . Find $d$ .
9. Find the sum of two-digit integers congruent to $3\pmod7$ .

Arithmetic Sequences Practice Problems — Hard