Mathematics, 23.10.2019 18:00 balancingmist1954

Find the common ratio and an explicit form in each of the following geometric sequences.
c.4 / 3,2 / 3,1 / 3,1 / 6 ,

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Answer from: brutalgitaffe

The common ratio is $\frac{1}{2}$ and the explicit form is $a_{n} =\frac{4}{3}r ^{n-1}$ for all n≥1.

Step-by-step explanation:

The general form of a geometric sequence is $a_{n}=ar^{n-1}$ , for all n≥1, where r≠0 is the common ratio and is the first term of the sequence.

From the problem we know the first term is a= $\frac{4}{3}$ , so we only need to calculate r. We know that $a_{2} =\frac{2}{3}$ is the second term of the sequence, so we can replace $a_{2}$ and a into our formula and find the value of r, so we have for n=2

$a_2=ar^{2-1}$

$\frac{2}{3} =\frac{4}{3} r$

$r=\frac{\frac{2}{3} }{\frac{4}{3} }=\frac{2}{4}=\frac{1}{2}$

so the explicit form is $a_{n}=a(\frac{1}{2})^{n-1}$ for all n≥1, where $a=\frac{4}{3}$

We can check the general formula by substituting the values into the equation.

$a_1=(\frac{4}{3} )(\frac{1}{2} ^{1-1})=(\frac{4}{3})(1)=\frac{4}{3} =a$

$a_2=(\frac{4}{3} )(\frac{1}{2} )^{2-1}=(\frac{4}{3})(\frac{1}{2} )=\frac{2}{3}$

$a_3=(\frac{4}{3} )(\frac{1}{2} )^{3-1}=(\frac{4}{3})(\frac{1}{2}^{2} )=(\frac{4}{3})(\frac{1}{4})=\frac{1}{3}$

$a_4=(\frac{4}{3} )(\frac{1}{2} )^{4-1}=(\frac{4}{3})(\frac{1}{2}^{3} )=(\frac{4}{3})(\frac{1}{8})=\frac{1}{6}$

We can conclude that the explicit form of the geometric sequence is $a_{n} =\frac{4}{3}r ^{n-1}$ for all n≥1, and the common ratio is $r=\frac{1}{2}$ .

Answer from: Quest

i got 10.28 but that rounds up to 10.3

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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Find the common ratio and an explicit form in each of the following geometric sequences.
c.4 /...