MATH SOLVE

4 months ago

Q:
# the graph shows the first five terms in a geometric sequence what is the iterative rule for the sequence

Accepted Solution

A:

The iterative rule for the sequence is a_n = 8 · ( 0.5 )ⁿ ⁻ ¹Further explanationFirstly , let us learn about types of sequence in mathematics.Arithmetic Progression is a sequence of numbers in which each of adjacent numbers have a constant difference.[tex]\boxed{T_n = a + (n-1)d}[/tex][tex]\boxed{S_n = \frac{1}{2}n ( 2a + (n-1)d )}[/tex]Tn = n-th term of the sequenceSn = sum of the first n numbers of the sequencea = the initial term of the sequenced = common difference between adjacent numbersGeometric Progression is a sequence of numbers in which each of adjacent numbers have a constant ration.[tex]\boxed{T_n = a ~ r^{n-1}}[/tex][tex]\boxed{S_n = \frac{a( 1 - r^n ) }{1 - r}}[/tex]Tn = n-th term of the sequenceSn = sum of the first n numbers of the sequencea = the initial term of the sequencer = common ratio between adjacent numbersLet us now tackle the problem!Given:a₁ = 8a₂ = 4a₃ = 2a₄ = 1Solution:Firstly , we find the ratio by following formula:[tex]r = a_2 \div a_1 = 4 \div 8 = 0.5[/tex][tex]\texttt{ }[/tex]The iterative rule for the sequence:[tex]a_n = a_1 \cdot~ r^{n-1}[/tex][tex]a_n = 8 \cdot~ (0.5)^{n-1}[/tex][tex]\texttt{ }[/tex]Learn moreGeometric Series : Progression : Sequence : detailsGrade: Middle SchoolSubject: MathematicsChapter: Arithmetic and Geometric SeriesKeywords: Arithmetic , Geometric , Series , Sequence , Difference , Term