91Ó°ÊÓ

Write each series as a sum of terms and then find the sum. $$ \sum_{i=1}^{3}\left(i^{2}+2\right) $$

Find the number of terms in each arithmetic sequence. A student incorrectly claimed that there are 100 terms in the arithmetic sequence $$ 2,4,6,8, \ldots, 100 $$ How many terms are there?

Write the first four terms of each binomial expansion. $$ (2 p-3 q)^{11} $$

Write the first four terms of each binomial expansion. $$ \left(x^{2}+y^{2}\right)^{15} $$

Find the indicated term of each binomial expansion. \((k-1)^{9} ;\) third term

A particular substance decays in such a way that it loses half its weight each day. In how many days will \(256 \mathrm{~g}\) of the substance be reduced to \(32 \mathrm{~g}\) ? How much of the substance is left after 10 days?

A seating section in a theater-in-the-round has 20 seats in the first row, 22 in the second row, 24 in the third row, and so on for 25 rows. How many seats are there in the last row? How many seats are there in the section?

The repeating decimal \(0.99999 \ldots\) can be written as the sum of the terms of a geometric sequence with \(a_{1}=0.9\) and \(r=0.1\) \(0.99999 \ldots=0.9+0.9(0.1)+0.9(0.1)^{2}+0.9(0.1)^{3}+0.9(0.1)^{4}+0.9(0.1)^{5}+\cdots\) Because \(|0.1|<1,\) this sum can be found from the formula \(S=\frac{a_{1}}{1-r} .\) Use this formula to find a more common way of writing the decimal \(0.99999 \ldots .\)