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Find \(a_{1}\) and \(d\) if \(a_{13}=13\) and \(a_{54}=54\)

Find the sum of the first 250 natural numbers.

Find the sum of the odd numbers from 1 to 99 inclusive.

Look for a pattern and then write an expression for the general term, or nth term, \(a_{n},\) of each sequence. Answers may vary. $$ 0,3,8,15,24, \dots $$

Many ancient Mayan pyramids were constructed over a span of several generations. Each layer of the pyramid has a stone perimeter, enclosing a layer of dirt or debris on which a structure once stood. One drawing of such a pyramid indicates that the perimeter of the bottom layer contains 36 stones, the next level up contains 32 stones, and so on, up to the top row, which contains 4 stones. How many stones are in the pyramid? CAN'T COPY THE IMAGE

Theaters are often built with more seats per row as the rows move toward the back. The Sanders Amphitheater has 20 seats in the first row, 22 in the second, 24 in the third, and so on, for 19 rows. How many seats are in the amphitheater? CAN'T COPY THE IMAGE

Find fraction notation for each infinite sum. (Each can be regarded as an infinite geometric series. $$ 0.7777 . . . $$

Find fraction notation for each infinite sum. Each can be regarded as an infinite geometric series. $$ 0.2222 \ldots $$

Use a graphing calculator to find \(S_{1}, S_{2}, S_{3},\) and \(S_{4}\) for each of the following sequences. $$ a_{n}=\frac{n+1}{n} $$

A frog is at the bottom of a 100 -ft well. With each jump, the frog climbs 4 ft, but then slips back 1 ft. How many jumps does it take for the frog to reach the top of the hole?