Arithmetic Sequences (Medium)

AMC 10/12 mid (≈Q9–16) — multi-step.

Arithmetic Sequences: core idea

Sums.

An arithmetic sequence has $a_3=11$ and $a_9=29$ . Find $a_1$ .

Hint: For sums, pair first and last terms or use $S_n=\frac n2(a_1+a_n)$ . Answer: 5.

1. Find the sum of the first $20$ terms of $4,7,10,\ldots$ .
2. In $2,5,8,\ldots$ , which term is $47$ ?
3. Find $d$ if $a_4=18$ and $a_{10}=48$ .
4. Find the sum of all multiples of $7$ from $7$ through $70$ .
5. An arithmetic sequence has $12$ terms, first term $3$ , last term $36$ . Find its sum.
6. Find $a_{15}$ if $a_5=8$ and $a_{20}=38$ .
7. How many terms of $100,94,88,\ldots$ are positive?
8. Find the sum $20+18+16+\cdots+2$ .
9. If the sum of the first $n$ positive integers is $78$ , find $n$ .