Determine Whether the Statement Is True or False

question 137

Multiple Choice

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. ​
If a₁, a₂, a₃, ... , an and b₁, b₂, b₃, ... , bn are geometric progressions, then a₁ b₁, a₂ b₂, a₃ b₃, ... , an bn is also a geometric progression.
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• A) It is false. If A₁, a₂, a₃, a₄, ... = 1, 3, 6, 9,.. B₁, b₂, b₃, b₄, ... = 2, 4, 8, 16,.. Are geometric progressions, then we can see that A₁ b₁, a₂ b₂, a₃ b₃, ... = 2, 12, 48, ... Is not a geometric progression.
• B) It is true. A geometric progression is completely determined if the first term and the common ratio are known. Thus, if A₁, a₂, a₃, ... , a_n , ... Is a geometric progression with the first term given by a and common ratio given by r, then by definition, A₁ = a A₂ = a₁ r = ar A₃ = a₂r = ar^{2 . . .} An = an_-1 = ar ^{n - 1} Thus, we see that the product of two nths terms of two geometric progressions is given by A _n b _n = a r ^{n - 1} b where a,b are the first terms of these progressions and r, r₁ are their common ratios respectively. Then we obtain that A₁, b₁, a₂, b₂, a₃, b₃, ... , an, bn is also geometric progression with first term ab and common ratio rr₁.

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Definitions:

Binding Contract

An agreement between parties that is legally enforceable by law, containing all essential elements of a contract.

Unilateral Contract

A contract in which only one party makes a promise or undertakes a performance, while the other party accepts by performing the specified act.

Oral Acceptance

The verbal agreement to the terms of a contract, which in some cases can be legally binding, depending on the nature of the contract and jurisdiction.

Battle Of The Forms

A situation in contract law where two parties form an agreement based on differing standard terms and conditions, typically through the exchange of forms like purchase orders and invoices.