Question

Please help me out some one. 2.) 2, 10, 50, 250, .... a. Identify each sequence as arithmetic or geometric, explain your answer. b. Find the next five terms. c. Write an explicit formula for the sequence. d. Write a recursive formula for the sequence.

Answer 1

`\begin{gathered} \text{ If we like to see if the sequence is arithmetic, the difference of two consecutive terms} \\ \text{ must remain constant, but} \\ 10-2=8 \\ 50-10=40 \end{gathered}`
`\begin{gathered} \text{ So we suspect the sequence is geometric, in that case, the division of two consecutive} \\ \text{ terms must be constant} \\ (10)/(2)=5 \\ (50)/(10)=5 \\ (250)/(50)=5 \\ \text{ Thus we indeed have a geometric sequence!} \end{gathered}`
`\begin{gathered} \text{and we find that the subsequent terms are obtained multiplying the previous one by 5 } \\ \text{ So the next five terms are:} \\ 250\cdot5=1250 \\ 1250\cdot5=6250 \\ 6250\cdot5=31250 \\ 31250\cdot5=156250 \\ 156250\cdot5=781250 \end{gathered}`
`\begin{gathered} \text{ Now, we find an explicit formula for the sequence by} \\ 10=2\cdot5 \\ 50=2\cdot5^2 \\ 250=2\cdot5^3 \\ So\text{ we have } \\ a_n=2\cdot5^n \end{gathered}`
`\begin{gathered} \text{ Since the next term is obtained by multiplying the previous one by 5, we have} \\ a_0=2, \\ a_(n+1)=5a_n\text{ for n}\ge0 \end{gathered}`