Determine the common ratio, and find the next three terms of each geometric sequence.
1. -¼, ½, -1, … .
Solution:
First, find the common ratio.

½÷(-¼)=-2
-1÷½=-2
The common ratio is -2. Multiply the third term by -2 to find the fourth term, and so on.
-1(-2)=2
2(-2)=-4
-4(-2)=8
Therefore, the next three terms are 2, -4, and 8.

2. 0.5, 0.75, 1.125, … .
Solution:
First, find the common ratio.

0.75÷0.5=1.5
1.125÷0.75=1.5
The common ratio is 1.5. Multiply the third term by 1.5 to find the fourth term, and so on.
1.125(1.5)=1.6875
1.6875(1.5)=2.53125
2.53125(1.5)=3.796875
Therefore, the next three terms are 1.6875, 2.53125, and 3.796875.

3. 8, 20, 50, … .
Solution:
First, find the common ratio.

20÷8 or 2.5
50÷20 or 2.5
The common ratio is 2.5. Multiply the third term by 2.5 to find the fourth term, and so on.
50(2.5)=125
125(2.5)=312.5
312.5(2.5)=781.25
Therefore, the next three terms are 125, 312.5, and 781 .25.

4. 2x, 10x, 50x, … .
Solution:
First, find the common ratio.

10x÷2x=5
50x÷10x=5
The common ratio is 5. Multiply the third term by 5 to find the fourth term, and so on.
5(50x)=250x
5(250x)=1250x
5(1250x)=6250x
Therefore, the next three terms are 250x, 1250x, and 6250x.

5. 64x, 16x, 4x, … .
Solution:
First, find the common ratio.

16x÷64x=¼
4x÷16x=¼
The common ratio is ¼. Multiply the third term by ¼ to find the fourth term, and so on.

Therefore, the next three terms are x, ¼x, and x/16.

6. ½, -⅜, 9/32, … .
Solution:
First, find the common ratio.

The common ratio is -¾. Multiply the third term by -¾ to find the fourth term, and so on.

Therefore, the next three terms are -27/128, 81/512, and=-243/2048.

7. x+5, 3x+15, 9x+45, … .
Solution:
First, find the common ratio.

The common ratio is 3. Multiply the third term by 3
to find the fourth term, and so on.
3(9x+45)=27x+135
3(27x+135)=81x+405
3(81x+405)=243x+1215
Therefore, the next three terms are 27x+135, 81x+405, and 243x+1215.

8. -9-y, 27+3y, -81-9y, …
Solution:
First, find the common ratio.
The common ratio is -3. Multiply the third term by -3 to find the fourth term, and so on.

-3(-81-9y)=243+27y
-3(243+27y)=-729-81y
-3(-729-81y)=2187+243y
Therefore, the next three terms are 243+27y, -729-81y, and 2187+243y.

Find the next three terms in each geometric sequence.
9. 10, 20, 40, 80, … .
Solution:

The common ratio is 2. Multiply each term by the common ratio to find the next three terms
80×2=160
160×2=320
320×2=640
The next three terms of the sequence are 160, 320, and 640.

10. 100, 50, 25, … .
Solution:
Calculate common ratio.

The common ratio is 0.5. Multiply each term by the common ratio to find the next three terms.
25×0.5=12.5
12.5×0.5=6.25
6.25×0.5=3.125
The next three terms of the sequence are 12.5, 6.25, and 3.125.

11. -7, 21, -63, … .
Solution:
Calculate the common ratio.

The common ratio is -3. Multiply each term by the common ratio to find the next three terms.
-63×(-3)=189
189×(-3)=-567
-567×(-3)=1701
The next three terms of the sequence are 189, -567, and 1701.

12. 2, 6, 18, 54, …
Solution:
Calculate the common ratio.

The common ratio is 3. Multiply each term by the common ratio to find the next three terms.
54×3=162
162×3=486
486×3=1458
The next three terms of the sequence are 162, 486, and 1458

13. -5, -10, -20, -40, …
Solution: Calculate the common ratio.

The common ratio is 2. Multiply each term by the common ratio to find the next three terms.
-40×2=-80
-80×2=-160
-160×2=-320
The next three terms of the sequence are -80, -160, and -320.

14. -3, 1.5, -0.75, 0.375, …
Solution:
Calculate the common ratio.

The common ratio is -0.5. Multiply each term by the common ratio to find the next three terms.
0.375×-0.5=-0.1875
-0.1875×-0.5=0.09375
0.09375×-0.5=-0.046875
The next three terms of the sequence are -0.1875, 0.09375, and -0.046875.

15. 1, 0.6, 0.36, 0.216, …
Solution:
Calculate the common ratio.

The common ratio is 0.6. Multiply each term by the common ratio to find the next three terms.
0.216×0.6=0.1296
0.1296×0.6=0.7776
0.7776×0.6=0.046656
The next three terms of the sequence are 0.1296, 0.07776, and 0.046656.

16. 4, 6, 9, 13.5, …
Solution:
Calculate the common ratio.

The common ratio is 1.5. Multiply each term by the common ratio to find the next three terms.
13.5×1.5=20.25
20.25×1.5=30.375
30.375×1.5=45.5625
The next three terms of the sequence are 20.25, 30.375, and 45.5625.

17. 2, -10, 50, … .
Solution:
Calculate the common ratio.

The common ratio is -5. Multiply each term by the common ratio to find the next three terms.
50×(-5)=-250
-250×(-5)=1250
1250×(-5)=-6250
The next three terms of the sequence are -250, 1250, and -6250.

18. 4, 12, 36, … .
Solution:
Calculate the common ratio.

The common ratio is 3. Multiply each term by the common ratio to find the next three terms.
36×3=108
108×3=324
324×3=972
The next three terms of the sequence are 108, 324, and 972.

19. -6, -42, -294, … .
Solution:
Calculate the common ratio.

The common ratio is 7. Multiply each term by the common ratio to find the next three terms.
-294×7=-2058
-2058×7=-14,406
-14,406×7=-100,842
The next three terms of the sequence are -2058, -14,406, and -100,842.

20. 1024, -128, 16, … .
Solution:
Calculate the common ratio.

The common ratio is -⅛. Multiply each term by the common ratio to find the next three terms.

The next three terms of the sequence are -2, ¼, and -1/32.

21. 4, -1, ¼, … .
Solution:
Calculate the common ratio.

The common ratio is -¼. Multiply each term by the common ratio to find the next three terms.

The next three terms of the sequence are -1/16, 1/64, and 1/256.

22. 1, -½, ¼, -⅛, …
Solution:
Calculate the common ratio.

The common ratio is (-½). Multiply each term by the common ratio to find the next three terms.

The next three terms of the sequence are 1/16, -1/32, and 1/64.

23. 36, 12, 4, … .
Solution:
Calculate the common ratio.

The common ratio is ⅓. Multiply each term by the common ratio to find the next three terms.

The next three terms of the sequence are 4/3, 4/9, and 4/27.

24. 400, 100, 25, … .
Solution:
Calculate the common ratio.

The common ratio is ¼. Multiply each term by the common ratio to find the next three terms.

The next three terms of the sequence are 25/16, 25/16, and 25/64.

Find the next three terms of each geometric sequence. Then graph the sequence.
25. 8, 16, 32, 64, … .
Solution:
Find the ratio of the consecutive terms.

Since the ratios are the same, the sequence is geometric.
To find the next term, multiply the previous term by 2.
64×2=128
128×2=256
256×2=512
The next three terms of the sequence are 128, 256, 512.
Graph the sequence.

26. 0.125, -0.5, 2, …
Solution:
Find the ratio of the consecutive terms.

Since the ratios are same, the sequence is geometric
To find the next term, multiply the previous term with -4.
2×(-4)=-8
-8×(-4)=32
32×(-4)=-128
The next three terms of the sequence are -8, 32, -128.
Graph the sequence.

27. ⅓, 1, 3, 9, …
Solution:
Find the ratio of the consecutive terms.

1÷⅓=3

3÷1=3

9÷3=3

Since the ratios are same, the sequence is geometric.
To find the next term, multiply the previous term by 3.

9×3=27
27×3=81
81×3=243
The next three terms of the sequence are 27, 81, 243.
Graph the sequence.

28. 1, 0.1, 0.01, 0.001, …
Solution:
Find the ratio of the consecutive terms.

Since the ratios are same, the sequence is geometric.
To find the next term, multiply the previous term by 0.1.
0.001×0.1=0.0001
0.0001×0.1=0.00001
0.00001×0.1=0.000001
The next three terms of the sequence are 0.0001, 0.00001, 0.000001.
Graph the sequence.

29. 8, 12, 18, 27, … .
Solution:
Find the ratio of the consecutive terms.

Since the ratios are the same. the sequence is geometric.
To find the next term. multiply the previous term by 3/2.

The next three terms of the sequence are 40.5, 60.75, 91.125.
Graph the sequence.

30. 250, 50, 10, 2, …
Solution:
Find the ratio of the consecutive terms.

Since the ratios are the same, the sequence is geometric.
To find the next term, multiply the previous term by ⅕.

The next three terms of the sequence are ⅖, 2/25, 2/125.
Graph the sequence.

31. 9, -3, 1, -⅓, …
Solution:
Find the ratio of the consecutive terms.

Since the ratios are the same, the sequence is geometric.
To find the next term, multiply the previous term by -⅓.

The next three terms of the sequence are 1/9, -1/27, 1/81.
Graph the sequence.

32.18, 12, 8, …
Solution:
Find the ratio of the consecutive terms.

Since the ratios are same, the sequence is geometric.
To find the next term, multiply the previous term by ⅔.

The next three terms of the sequence are 16/3, 32/9, 64/27.
Graph the sequence.

33. 64, 48, 36, …
Solution: Find the ratio of the consecutive terms.

Since the ratios are same, the sequence is geometric.
To find the next term, multiply the previous term by ¾.

The next three terms of the sequence are 27, 81/4, 243/16.
Graph the sequence.

34. 81, 108, 144, …
Solution:
Find the ratio of the consecutive terms.

Since the ratios are same, the sequence is geometric.
To find the next term, multiply the previous term by 4/3.

The next three terms of the sequence are 192, 256, 1024/3.
Graph the sequence.

35. What are the next three terms of the following sequence?
4, 20, 100, 500, …
Solution:

500⋅5=2500
2500⋅5=12,500
12,500⋅5=62,500
The next three terms are 2,500, 12,500, and 62,500.

36. Find r and the next four terms of the G.P.
-3, 1, -⅓, 1/9, …
Solution:
Here a=-3, ar=1
So that

The next four terms of geometric sequence are

37. (Tourist Attractions) To prove that objects of different weights fall at the same rate, Marlene dropped two objects with different weights from the Leaning Tower of Pisa in Italy. The objects hit the ground at the same time. When an object is dropped from a tall building, it falls about 16 feet in the first second, 48 feet in the second second, and 80 feet in the third second, regardless of its weight. If this pattern continues, how many feet would an object fall in the sixth second?
Solution:
In this sequence, a₁=16, a₂=48, and a₃=80.
Find the common difference.

48-16=32
80-48=32
Find a_6.
80+32=112
112+32=144
144+32=176
Therefore, the object will fall 176 feet in the sixth second.