Given:
The series is
[tex]2x+2x^2+2x^3+...+2x^n[/tex]
To find:
Whether the given series is arithmetic, geometric, both, or neither.
Solution:
We have,
[tex]2x+2x^2+2x^3+...+2x^n[/tex]
Difference between consecutive terms are:
[tex]d_1=a_2-a_1[/tex]
[tex]d_1=2x^2-2x[/tex]
[tex]d_1=2x(x-1)[/tex]
And,
[tex]d_2=a_3-a_2[/tex]
[tex]d_2=2x^3-2x^2[/tex]
[tex]d_2=2x^2(x-1)[/tex]
Here, [tex]d_1\neq d_2[/tex].
So, the given series is not an arithmetic series.
Ratio between consecutive terms are:
[tex]r_1=\dfrac{a_2}{a_1}[/tex]
[tex]r_1=\dfrac{2x^2}{2x}[/tex]
[tex]r_1=x[/tex]
And,
[tex]r_2=\dfrac{a_3}{a_2}[/tex]
[tex]r_2=\dfrac{2x^3}{2x^2}[/tex]
[tex]r_2=x[/tex]
Similarly upto last pair of consecutive terms.
[tex]r_{n-1}=\dfrac{a_n}{a_{n-1}}[/tex]
[tex]r_{n-1}=\dfrac{2x^n}{2x^{n-1}}[/tex]
[tex]r_{n-1}=x[/tex]
Since, [tex]r_1=r_2=...=r_{n-1}[/tex], so the given series have a common ratio.
Therefore, it is a geometric series.
