91Ó°ÊÓ

Match the expression with the most appropriate expression from the column on the right. ____ \(\sum_{k=1}^{4} k^{2}\) a) \(-1+1+(-1)+1\) b) \(a_{2}=25\) c) \(a_{2}=8\) d) \(\sum_{k=1}^{4} 5 k\) e) \(S_{3}\) f) \(1+4+9+16\)

Simplify. $$9 !$$

Find the common ratio for each geometric sequence. $$75,15,3, \frac{3}{5}, \dots$$

The nth term of a sequence is given. Find the first 4 terms; the 10 th term, \(a_{10} ;\) and the 15 th term, \(a_{15},\) of the sequence. $$a_{n}=3 n-1$$

Find the first term and the common difference. Find \(a_{1}\) and \(d\) if \(a_{12}=24\) and \(a_{25}=50\)

It is said that as a young child, the mathematician Karl F. Gauss \((1777-1855)\) was able to compute the sum \(1+2+3+\cdots+100\) very quickly in his head. Explain how Gauss might have done this and present a formula for the sum of the first \(n\) natural numbers. (Hint: \(1+99=100 .)\)

Find the indicated term for each binomial expression. Maya claims that she can calculate mentally the first two terms and the last two terms of the expan sion of \((a+b)^{n}\) for any whole number \(n .\) How do you think she does this?

Write out and evaluate each sum. $$ \sum_{k=0}^{5}\left(k^{2}-2 k+3\right) $$

Form 1 of the binomial theorem can be proved using form 2 of the binomial theorem. The key step in that proof is showing that the coefficients inside Pascal's triangle are found by adding the two terms above. Prove this fact by showing that $$\left(\begin{array}{l}{n} \\\\{r}\end{array}\right)=\left(\begin{array}{c}{n-1} \\ {r-1}\end{array}\right)+\left(\begin{array}{c}{n-1} \\\\{r}\end{array}\right)$$

Rewrite each sum using sigma notation. Answers may vary. $$ 9-16+25+\dots+(-1)^{n+1} n^{2} $$