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### High School Mathematics - 23.5 Geometric Sequence

Geometric Sequence
• A sequence such as 1,2,4,8,........1024 or 2,6,18,54...........118098 which has a constant ratio between terms.
• The first term is written as a1,
• The second term is written as a2,
• The third term is written as a3,
• The fourth term is written as a4,
• The fifth term is written as a5, ....
• The tenth terms is written as a10 and
• The ratio as r
Examples:
• 1,2,4,8,........1024 = 1x20 , 1x21 , 1x22 , 1x23,........
• 2,4,8,16,32........1024 = 2x20 , 2x21 , 2x22 , 2x23,........
• 1,3,9,27,........ = 1x30 , 1x31 , 1x23 , 1x24,........
• 4,16,64........= 1x41 , 1x42 , 1x43,........
• 7,21,63........= 7x30 , 7x31 , 7x32,........
• 10,100,1000,10000,....... = 1x101, 1x102, 1x103, 1x104.............
• 2,4,8,........1024

The first term is a1, the 5th term as a5, the 10th terms as a10 and the ratio as r.
a1r0
a2 = a1 r1
a3 = a2 r1 = (a1 r1) r1 = a1 r2
a4 = a3 r1 = (a2r1) r1 = a1 r3
a5 = a1 r4
a6 = a1r5
a7 = a1 r6
........
an = a1rn-1

The formula to find the nth term in geometric series is
 a1 rn-1

Example:
Power notation:
• 1=100, 10 = 101, 100 = 102, 1000 = 103, 10000 = 104............

• 1 = 20, 2=21, 4=22, 8=23, ............1024=210
Let the first term be a1
the 5th term be a5
the 10th term be a10
ratio be r
a1 = 20
a5 = 24
a10 = 29
r = 2

• 1 = 30, 3=31, 9=32, 27=33, ............59049=310

2 = 2x30
6 = 2x31
18 = 2x32
54 = 2x33
...........
118098=2x310
 Term Number 1 2 3 4 ..... n Term of Geometric Series a1 a2 a3 a4 ..... an Term of Geometric Series a1 r0 a1r1 a1r2 a1r3 ..... a1 rn-1 Example 2x30 2x31 2x32 2x33 ........... 2x3n-1

More Examples:
 Geometric Sequence Common Ratio 3, 1, 1/3, 1/32, 1/33,. . . . 1/3 2, -2Ö3, 6, -6Ö3, 18 . . . . -Ö3

Directions: Find r, a1, a5, a101 for the following arithmetic sequence. Also write at least 5 examples of your own.
 Q 1: What is the a10 for the sequence: 3,6,12,24,48,...........210015361400 Q 2: What is the ratio for the sequence: 1/2, -1, 2, -4, 8, 16, .........2-21/2 Q 3: What is the ratio for the sequence: 16 , -8 , 4 , -2 , 1 , ... -2-1/22 Q 4: What is the a5 for the sequence: 1/2, -1, 2, -4, 8, 16, .........-482 Q 5: What is the first term for the sequence: 3,6,12,24,48,...........3126 Q 6: What is the a10 for the sequence: 1/2, -1, 2, -4, 8, 16, .........200312256 Q 7: What is the ratio for the sequence: 2, 6, 18, 54, 162, . . . 236 Q 8: What is the a5 for the sequence: 3,6,12,24,48,...........482412 Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!