Question

Consider the following sequence: 7,14,28,56,... find a19

Answer 1

We have the sequence:

`\begin{gathered} a_1=7, \\ a_2=14, \\ a_3=28, \\ a_4=56, \\ \ldots \end{gathered}`

We rewrite the sequence as:

`\begin{gathered} a_1=7\cdot1=7\cdot2^0=7\cdot2^(1-1), \\ a_2=7\cdot2=7\cdot2^1=7\cdot2^(2-1), \\ a_3=7\cdot4=7\cdot2^2=7\cdot2^(3-1), \\ a_4=7\cdot8=7\cdot2^3=7\cdot2^(4-1), \\ \ldots \end{gathered}`

From the previous equations, we see that the general term is given by:

`a_n=7\cdot2^(n-1).`

Replacing n = 19, we get:

`a_(19)=7\cdot2^(19-1)=7\cdot2^(18)=7\cdot262144=1835008.`

Answer

`a_(19)=7\cdot2^(18)=1835008`