Mathematics, 15.12.2020 14:00 Delgadojacky0206

Identify the following series as arithmetic, geometric, both, or neither $2x + 2 {x}^{2} + 2 {x}^{3} + ... + 2 {x}^{n}$

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Answer from: Paris4life

Given:

The series is

2x+2x^2+2x^3+...+2x^n

To find:

Whether the given series is arithmetic, geometric, both, or neither.

Solution:

We have,

2x+2x^2+2x^3+...+2x^n

Difference between consecutive terms are:

d_1=a_2-a_1

d_1=2x^2-2x

d_1=2x(x-1)

And,

d_2=a_3-a_2

d_2=2x^3-2x^2

d_2=2x^2(x-1)

Here, $d_1\neq d_2$ .

So, the given series is not an arithmetic series.

Ratio between consecutive terms are:

$r_1=\dfrac{a_2}{a_1}$

$r_1=\dfrac{2x^2}{2x}$

r_1=x

And,

$r_2=\dfrac{a_3}{a_2}$

$r_2=\dfrac{2x^3}{2x^2}$

r_2=x

Similarly upto last pair of consecutive terms.

$r_{n-1}=\dfrac{a_n}{a_{n-1}}$

$r_{n-1}=\dfrac{2x^n}{2x^{n-1}}$

$r_{n-1}=x$

Since, $r_1=r_2=...=r_{n-1}$ , so the given series have a common ratio.

Therefore, it is a geometric series.

Answer from: Quest

Then just do it (lol)

Answer from: Quest

Answer: question no. 1 -a. yes, the sides are in the ratio 2 : 5.question no. 2 -b. 59°question no. 3 -c. angles x and y are remote interior angles.question no. 4 -d. 90°, 45° and 45°question no. 5 -b. mn and qrstep-by-step explanation: question no. 1 -rs : xu = 3 : 7.5 = 2 : 5rt : x? = 6.4 : 16 = 2 : 5st : u? = 6 : 15 = 2 : 5as the ratios of all sides are equivalent, the triangles must be congruent and the sides must be proportional.therefore, the answer is: a. yes, the sides are in the ratio 2 : 5.question no. 2 -in the second diagram, angle b is the only letter not shown. therefore, the number must be equivalent to angle b.if we substitute the value of angle b as being 22° in the first diagram, we can now use the angle sum of a triangle to obtain the value of angle a.angle sum of a triangle = 180°therefore: let angle a = aa + 99 + 22 = 180a = 180 - 99 - 22a = 59°therefore, the answer is: b. 59°question no. 3 -angles x and y don't share the vertex of the triangle with the exterior angle w. remote interior angles don't share a vertex of a triangle with an exterior angle. this means that angles x and y are remote interior angles.therefore, the answer is: c. angles x and y are remote interior angles.question no. 4 -the sum of all interior angles of a triangle must be equivalent to 180°.90°, 45° and 45° are the only set of angle measures that are equivalent to 180°.therefore, the answer is: d. 90°, 45° and 45°question no. 5 -the only pair of sides stated in the options which are corresponding are mn and qr.therefore, the answer is: b. mn and qr.

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