Mod 8 Honors Assignment - Sequences and Series
Connecting Arithmetic and Geometric Sequences
Example: Three numbers form a geometric sequence whose common ratio
is 3. If the first is decreased by 1, the second decreased by 3, and the third
reduced to 2 less than half its value, the resulting three numbers form an
arithmetic sequence. Determine the original three numbers.
Step 1: Represent the 3 numbers that form a
geometric sequence:

x, 3x, 9x

Step 2: Represent the 3 numbers that form an
arithmetic sequence:

x − 1, 3x − 3,
9
2
2
x −

Step 3: Apply the difference property of an
arithmetic sequence:

( ) ( )
9
3 3 1
2

x


− − − = 


Step 4: Solve for x x = 6
The original numbers are

x x x = = = 6, 3 18, 9 54

Show and Explain all work for each problem
1. Three numbers form a geometric sequence
whose common ratio is 0.5. If the first is reduced
to 10 more than one quarter its value, the second
decreased by 10, and the third increased by 10
more than twice its value, the resulting three
numbers form an arithmetic sequence. Determine
the original three numbers. 30 points