## Thinking Mathematically (6th Edition)

The given sequence is an arithmetic sequence and the next two terms of the given sequence are$-28\ \text{and}\ -35$.
To check is a sequence is an arithmetic sequence, see if the differences between two consecutive terms are equal i.e. check if${{a}_{n+1}}-{{a}_{n}}=d\ \$ for all n element of N, here $d$ is the common difference. To check if a sequence is a geometric sequence, see if the ratio between two consecutive terms is equal i.e. check if $\frac{{{a}_{n+1}}}{{{a}_{n}}}=r\$for all n element of N here $r$ is the common ratio. When$n=1$, for the given sequence, \begin{align} & d={{a}_{2}}-{{a}_{1}} \\ & =-7-0 \\ & =-7 \end{align} And, \begin{align} & r=\frac{{{a}_{2}}}{{{a}_{1}}} \\ & =\frac{-7}{0} \\ & =\text{Not defined} \end{align} When$n=2$, for the given sequence, \begin{align} & d={{a}_{3}}-{{a}_{2}} \\ & =-14-\left( -7 \right) \\ & =-7 \end{align} And, \begin{align} & r=\frac{{{a}_{3}}}{{{a}_{2}}} \\ & =\frac{-14}{-7} \\ & =2 \end{align} When$n=3$, for the given sequence, \begin{align} & d={{a}_{4}}-{{a}_{3}} \\ & =-21-\left( -14 \right) \\ & =-7 \end{align} And, \begin{align} & r=\frac{{{a}_{4}}}{{{a}_{3}}} \\ & =\frac{-21}{-14} \\ & =1.5 \end{align} Since, $d=-7\ \ \forall n=1,2,3$and $r$is not equal for all n element of N it implies the given sequence is an arithmetic sequence. Use the formula ${{a}_{n}}={{a}_{n-1}}+d$to find the next term of the arithmetic sequence. Put $n=5$in${{a}_{n}}={{a}_{n-1}}+d$to find the fifth term of the given arithmetic sequence as follows: \begin{align} & {{a}_{5}}={{a}_{5-1}}+d \\ & ={{a}_{4}}+d \\ & =-21-7 \\ & =-28 \end{align} Put $n=6$in${{a}_{n}}={{a}_{n-1}}+d$to find the sixth term of the given arithmetic sequence as follows: \begin{align} & {{a}_{6}}={{a}_{6-1}}+d \\ & ={{a}_{5}}+d \\ & =-28-7 \\ & =-35 \end{align} The given sequence is an arithmetic sequence and the next two terms of the given sequence are$-28\ \text{and}\ -35$.