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.

Calculate probabilities and parameters (mean, variance, expected value) for various probability distributions.

Distinguish between characteristics of continuous and discrete probability distributions.

Interpret the effects of changes in the parameters of a probability distribution on its shape and probabilities.

Utilize standard normal distribution properties and calculations, including z-values and probabilities.

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Definitions:

Deduction

A logical process in which a conclusion is based on the concordance of multiple premises that are assumed to be true.

Discipline

A method of training people to obey rules or a code of behavior, using punishment to correct disobedience.

Gender Constancy

The concept that one’s gender remains the same despite changes in appearance or behavior.

Kohlberg

Refers to Lawrence Kohlberg, who developed a theory on the stages of moral development in individuals, highlighting the progression from basic, self-centered ethics to higher-order thinking involving societal norms and justice.