Question

In two or more complete sentences, explain whether the sequence is finite or infinite. Describe the pattern in the sequence if it exists, and if possible find the sixth term. 2a, 2a2b, 2a3b2, 2a4b3. . .

Answer 1

This is infinite since the sequence can go on forever. The sequence increases by 1b every time and the end number is increased 1 every time as well.
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Answer 2

Answer:

a). it is an Infinite Sequence.

b). This sequence is Geometric Progression.

c).  $6th\,term\,of\,this\,GP\,=\,2a^6b^5$                              

Step-by-step explanation:

Given: A Sequence 2a , 2a²b , 2a³b² , $2a^4b^3$

To find: a). Given Sequence is finite or infinite

             b). Pattern of the sequence

             c). Sixth term

a). Since, there is no detail about no. of terms of the sequence.

     Therefore, It implies it is an Infinite Sequence.

b). Given 1st term = 2a  ,  2nd term = 2a²b , 3rd term = 2a³b²

    First we find the difference of 1st and 2nd term

    d = 2a²b - 2a  ⇒  d = 2a (ab - 1)

    Now, Difference of 2nd and 3rd term,

    d = 2a³b² - 2a²b = 2a²b (ab - 1)

So, This Sequence is not a Arthematic Progression.

Now, we find the ratio of 2nd term to 1st term,

$r\:=\:\frac{2a^2b}{2a}\,=\,ab$

Ratio of 3rd term to 2nd term,

$r\:=\:\frac{2a^3b^2}{2a^2b}\,=\,ab$

Since, the ratio is common.

This sequence is Geometric Progression.

So, Pattern is next term of this sequence has 1 more in  power of a and b

⇒ $nth\,term\,of\,this\,GP\,=\,2a^{n}b^{n-1}$

c). $6th\,term\,of\,this\,GP\,=\,2a^{6}b^{6-1}$

    $6th\,term\,of\,this\,GP\,=\,2a^6b^5$

Therfore, a). it is an Infinite Sequence.

               b). This sequence is Geometric Progression.

              c).  $6th\,term\,of\,this\,GP\,=\,2a^6b^5$