91Ó°ÊÓ

Simplify. $$ \left(a_{1}+3 d\right)+d $$

Simplify. $$ \left(a_{1}+5 d\right)+\left(a_{n}-5 d\right) $$

Simplify. $$ \left(a_{1}+a_{n}\right)+\left(a_{1}+a_{n}\right)+\left(a_{1}+a_{n}\right) $$

Simplify. $$ \left(a_{1}+8 d\right)-\left(a_{1}+7 d\right) $$

Explain why the equation $$ \sum_{k=1}^{n}\left(a_{k}+b_{k}\right)=\sum_{k=1}^{n} a_{k}+\sum_{k=1}^{n} b_{k} $$ is true for any positive integer \(n .\) What laws are used to justify this result?

Some sequences are given by a recursive definition. The value of the first term, \(a_{1},\) is given, and then we are told how to find any subsequent term from the term preceding it. Find the first six terms of each of the following recursively defined sequences. $$ a_{1}=1, a_{n+1}=5 a_{n}-2 $$

Some sequences are given by a recursive definition. The value of the first term, \(a_{1},\) is given, and then we are told how to find any subsequent term from the term preceding it. Find the first six terms of each of the following recursively defined sequences. $$ a_{1}=0, a_{n+1}=a_{n}^{2}+3 $$

A single cell of bacterium divides into two every 15 min. Suppose that the same rate of division is maintained for 4 hr. Give a sequence that lists the number of cells after successive \(15-\min\) periods.

Find the first five terms of each sequence. Then find \(S_{5}\). $$ a_{n}=\frac{1}{2^{n}} \log 1000^{n} $$

Find the first five terms of each sequence. Then find \(S_{5}\). $$ a_{n}=i^{n}, i=\sqrt{-1} $$