Activity 5:Missing You
A. Find the missing terms in each geometric sequence.
1. 3,12,48,__ , __
2. __,__,32,64,128
3.120,60,30,__,__,__
4. 5,__,20,40,__,__
5.__,4,12,40,__,__
6.-2,__,__,-16,-32,-64
7. 256,__,__,-32,16,...
8. 27,9,__,__,1/3
9.1/4,__,__,__,64,256
B. Insert 3 terms between 2 and 32 of geometric sequence.


Sagot :

Answer:

Step-by-step explanation:

Geometric Sequence:

A geometric sequence is a series of number that follows an order.

It follows a rule such as the below:

Xₙ = ar⁽ⁿ⁻¹⁾

Where:

n = is the nth term

a = is the first terms

r = is the common ratio

Now that we have the formula for the geometric sequence, we can now calculate the following geometric sequences:

1) 3,12,48,__ , __

First we need to determine what is the common ratio "r"

looking at the sequence, the common ration is 4 because

3 = first digit

3 x 4 = 12 (the second digit)

12 x 4 = 48 (the third digit)

Hence,

r = 4

The missing digit in this sequence is the 4th and 5th digit, thus using the formula given above:

For the 4th digit:

Xₙ = ar⁽ⁿ⁻¹⁾

X₄ = 3(4)⁽⁴⁻¹⁾

X₄ = 3(4)⁽³⁾

X₄ = 3(64)

X₄ = 192

For the 5th digit:

Xₙ = ar⁽ⁿ⁻¹⁾

X₅ = 3(5)⁽⁵⁻¹⁾

X₅ = 3(4)⁽⁴⁾

X₄ = 3(256)

X₄ = 768

Therefore the geometric sequence is:

3, 12, 48, 192 , 768

2) __,__,32,64,128

First find out "r"

From the 3rd to 5th digit:

the common ratio is 2

(because 128 ÷ 64 = 2, similarly 64 ÷ 32 = 2) thus, r = 2

For the 1st digit "a":

We first need to determine a from the already given sequence, let's say the 3rd sequence:

32 = a(2)⁽³⁻¹⁾

32 = a(2)⁽²⁾

32 = a(4)

a = 32 ÷ 4

Thus the 1st digit is:

a = 8

For the 2nd digit

Xₙ = ar⁽ⁿ⁻¹⁾

X₂ = 8(2)⁽²⁻¹⁾

X₂ = 3(4)⁽¹⁾

X₂ = 3(4)

X₂ = 12

Therefore the geometric sequence is:

8,12,32,64,128

3) 120,60,30,__,__,__

Find r:

From the standard formula for geometric sequence:

Xₙ = ar⁽ⁿ⁻¹⁾

Lets use the 2nd digit from the sequence, 60

60 = 120r⁽²⁻¹⁾

60 = 120r⁽¹⁾

[tex]\frac{60}{120}[/tex] = r

r = [tex]\frac{1}{2}[/tex] or 0.5

Thus finding the 4th, 5th and 6th digits would be:

X₄ = 120(0.5)⁽⁴⁻¹⁾

X₄ = 120(0.5)⁽³⁾

X₄ = 120(0.125)

X₄ = 15 → 4th digit

--

X₅ = 120(0.5)⁽⁵⁻¹⁾

X₅ = 120(0.5)⁽⁴⁾

X₅ = 120(0.0625)

X₅ = 7.5 → 5th digit

--

X₆ = 120(0.5)⁽⁶⁻¹⁾

X₆ = 120(0.5)⁽⁵⁾

X₅ = 120(0.03125)

X₆ = 3.75 → 5th digit

Thus the geometric sequence would be:

120 ,60 ,30 ,15, 7.5 , 3.75

For items 4 to 9 geometric sequence, I will not be showing the full solution as the above solutions would be enough for you to be able to solve the remaining problems, I will only give the common ratio "r" and the complete geometric sequences:

4) 5,__,20,40,__,__

Ans:

r = 2

5, 10, 20, 40, 80, 120

5)__,4,12,40,__,__

Lets find r:

Xₙ = ar⁽ⁿ⁻¹⁾

4 = ar⁽²⁻¹⁾

4 = ar⁽¹⁾

4=ar

thus;

a = [tex]\frac{4}{r}[/tex]

For the third term

12 = [tex]\frac{4}{r}[/tex](r)⁽³⁻¹⁾

12 =  [tex]\frac{4}{r}[/tex] r²

in this case the donominator "r" will be cancelled.

12= 4r

thus r = 3

To find the first term lets use the 3rd digit from the sequence

12 = a(3)⁽³⁻¹⁾

12 = a(3)²

12 = a(9)

thus

a = [tex]\frac{12}{9}[/tex]

or

a= 1 [tex]\frac{1}{3}[/tex] → 1.3333

For the 5th term:

X₅ = 1.333(3)⁽⁵⁻¹⁾

X₅ = 1.333 (3)⁽⁴⁾

X₅ = 1.333(81)

X₅ = 108

For the 6th term:

X₆ = 1.333(3)⁽⁶⁻¹⁾

X₆ = 1.333 (3)⁽⁵⁾

X₆ = 1.333(243)

X₆ = 324

Thus the geometric sequence would be:

1 1/3 ,4 ,12 ,40,108,324

6)-2,__,__,-16,-32,-64

Ans:

r = 2

-2,-4, -8 ,-16,-32,-64

7) 256,__,__,-32,16

Ans:

r = 0.5

256,-128,64,-32,16

8) 27,9,__,__,1/3

Ans:

r = 9/27

27 , 9 , 3 , 1 , 1/3

9)1/4,__,__,__,64,256

Ans:

r = 4

1/4 , 1 , 4 , 16 ,64,256

B. Insert 3 terms between 2 and 32 of geometric sequence.

Thus,

2, __, __, __, 32

To find r,

Xₙ = ar⁽ⁿ⁻¹⁾

32 = 2r⁽⁵⁻¹⁾

32 = 2r⁴

r⁴ = 32/2

r = [tex]\sqrt[4]{16}[/tex]

r = 2

Xₙ = ar⁽ⁿ⁻¹⁾

X₂ = 2(2)⁽²⁻¹⁾

X₂ = 2(2)⁽¹⁾

X₂ = 2(2)

X₂ = 4

__

Xₙ = ar⁽ⁿ⁻¹⁾

X₃ = 2(2)⁽³⁻¹⁾

X₃ = 2(2)⁽²⁾

X₃ = 2(4)

X₃ = 8

__

Xₙ = ar⁽ⁿ⁻¹⁾

X₄ = 2(2)⁽⁴⁻¹⁾

X₃ = 2(2)⁽³⁾

X₃ = 2(8)

X₃ = 16

Thus the geometric sequence would be:

2, 4 , 8 , 16 , 32

For more information regarding geometric sequence visit the link below:

https://brainly.ph/question/1583842

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