Write the first five terms of the geometric sequence. Find the common ratio and write the nth...

Given:

• The first term is {eq}{a_1} = 200 {/eq}.

• The general term is {eq}{a_{k + 1}} = 0.1{a_k} {/eq}.

The objective is to find the first five terms, the common ratio and the general term of the sequence.

The first term is given as {eq}{a_1} = 200 {/eq}.

Find the second term by substituting {eq}k = 1 {/eq} in the general term {eq}{a_{k + 1}} = 0.1{a_k} {/eq}, and by using {eq}{a_1} = 200 {/eq}.

{eq}\begin{align*} {a_{1 + 1}} &= 0.1{a_1}\\ {a_2} &= 0.1\left( {200} \right)\\ {a_2} &= 20 \end{align*} {/eq}

Thus, the second term is 20.

Find the third term by substituting {eq}k = 2 {/eq} in the general term {eq}{a_{k + 1}} = 0.1{a_k} {/eq}, and by using {eq}{a_2} = 20 {/eq}.

{eq}\begin{align*} {a_{2 + 1}} &= 0.1{a_2}\\ {a_3} &= 0.1\left( {20} \right)\\ {a_3} &= 2 \end{align*} {/eq}

Thus, the third term is 2.

Find the fourth term by substituting {eq}k = 3 {/eq} in the general term {eq}{a_{k + 1}} = 0.1{a_k} {/eq}, and by using {eq}{a_3} = 2 {/eq}.

{eq}\begin{align*} {a_{3 + 1}}& = 0.1{a_3}\\ {a_4}& = 0.1\left( 2 \right)\\ {a_4}& = 0.2 \end{align*} {/eq}

Thus, the fourth term is 0.2.

Find the fifth term by substituting {eq}k = 4 {/eq} in the general term {eq}{a_{k + 1}} = 0.1{a_k} {/eq}, and by using {eq}{a_4} = 0.2 {/eq}.

{eq}\begin{align*} {a_{4 + 1}}& = 0.1{a_4}\\ {a_5}& = 0.1\left( {0.2} \right)\\ {a_5}& = 0.02 \end{align*} {/eq}

Thus, the fifth term is 0.02.

Find the common ratio as follows:

{eq}\begin{align*} r &= \dfrac{{{a_2}}}{{{a_1}}}\\ r &= \dfrac{{20}}{{200}}\\ r &= 0.1 \end{align*} {/eq}

Thus, the common ratio is {eq}r = 0.1 {/eq}.

Find the {eq}{n^{{\rm{th}}}} {/eq} term of the geometric sequence as follows:

{eq}\begin{align*} {a_n} &= {a_1}{r^{n - 1}}\\ {a_n} &= \left( {200} \right){\left( {0.1} \right)^{n - 1}}\\ {a_n} &= \left( {200} \right){\left( {0.1} \right)^n} \cdot \left( {10} \right)\\ {a_n} &= 2000{\left( {0.1} \right)^n} \end{align*} {/eq}

Thus, the {eq}{n^{{\rm{th}}}} {/eq} term is {eq}{a_n} = 2000{\left( {0.1} \right)^n} {/eq}.

Thus, the first five terms are {eq}200,20,2,0.2,0.02 {/eq}, the common ratio is 0.1, and the {eq}{n^{{\rm{th}}}} {/eq} term is {eq}{a_n} = 2000{\left( {0.1} \right)^n} {/eq}.