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Davion Harding

Davion Harding

Answered question

2022-06-20

If a 1 , a 2 , a 3 , and b 1 , b 2 , b 3 , . . . are geometric sequences, show that a 1 b 1 , a 2 b 2 , a 3 b 3 , is also a geometric sequence.

Answer & Explanation

Elianna Douglas

Elianna Douglas

Beginner2022-06-21Added 23 answers

Use the geometric sequence property
If a, b and c are geometric sequence then b 2 = a × c
a 2 2 = a 1 × a 3 (1)
b 2 2 = b 1 × b 3 (2)
Now, the sequence a 1 b 1 , a 2 b 2 , a 3 b 3 , . . . in G.P
If ( a 2 b 2 ) 2 = a 1 b 1 × a 3 b 3
Consider the left-hand side ( a 2 b 2 ) 2
Simplify
( a 2 b 2 ) 2 = a 2 2 × b 2 2 (3)
Substitute a 2 2 = a 1 × a 3 and b 2 2 = b 1 × b 3
We get ( a 2 b 2 ) 2 = a 1 × a 3 × b 1 × b 3
Rewrite the equation
( a 2 b 2 ) 2 = ( a ) 1 b 1 a 3 b 3
Thus, a 1 b 1 , a 2 b 2 , a 3 b 3 , . . . is a geometric sequence.

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