Which recursive sequence would produce the sequence 6,-33,162,dots ? a_(1)=6 and a_(n)=3a_(n-1)-6 a_(1)=6 and a_(n)=-5a_(n-1)-3 a_(1)=6 and a_(n)=-6a_(n-1)+3 a_(1)=6 and a_(n)=-3a_(n-1)-5

Test First Option: Test the first option. Given: a_(1)=6 and a_(n)=3a_(n-1)-6 Calculate a_(2) using the given formula: a_(2)=3a_(1)-6=3*6-6=18-6=12 Check if this matches the sequence: 6, -33, 162, ...Incorrect First Option: Since a_(2) does not match the second term of the sequence, which is -33, the first option is incorrect.Test Second Option: Test the second option. Given: a_(1)=6 and a_(n)=-5a_(n-1)-3 Calculate a_(2) using the given formula: a_(2)=-5a_(1)-3=-5*6-3=-30-3=-33 Check if this matches the sequence: 6, -33, 162, ...Verify Second Option: Since a_(2) matches the second term of the sequence, continue to calculate a_(3) to verify the pattern. Calculate a_(3) using the given formula: a_(3)=-5a_(2)-3=-5*(-33)-3=165-3=162 Check if this matches the sequence: 6, -33, 162, ...Correct Second Option: Since a_(3) matches the third term of the sequence, the second option correctly produces the sequence.

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Which recursive sequence would produce the sequence
6,-33,162,dots ?

a_(1)=6 and
a_(n)=3a_(n-1)-6

a_(1)=6 and
a_(n)=-5a_(n-1)-3

a_(1)=6 and
a_(n)=-6a_(n-1)+3

a_(1)=6 and
a_(n)=-3a_(n-1)-5

Which recursive sequence would produce the sequence $6,-33,162, \ldots$ ? $\newline$ $a_{1}=6$ and $a_{n}=3 a_{n-1}-6$ $\newline$ $a_{1}=6$ and $a_{n}=-5 a_{n-1}-3$ $\newline$ $a_{1}=6$ and $a_{n}=-6 a_{n-1}+3$ $\newline$ $a_{1}=6$ and $a_{n}=-3 a_{n-1}-5$

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Math Problems

Algebra 1

Identify a sequence as explicit or recursive

Full solution

Q. Which recursive sequence would produce the sequence

6,-33,162, \ldots

\newline

a_{1}=6

and

a_{n}=3 a_{n-1}-6

\newline

a_{1}=6

and

a_{n}=-5 a_{n-1}-3

\newline

a_{1}=6

and

a_{n}=-6 a_{n-1}+3

\newline

a_{1}=6

and

a_{n}=-3 a_{n-1}-5

Test First Option: Test the first option. $\newline$ Given: $a_{1}=6$ and $a_{n}=3a_{n-1}-6$ $\newline$ Calculate $a_{2}$ using the given formula: $a_{2}=3a_{1}-6=3\times 6-6=18-6=12$ $\newline$ Check if this matches the sequence: $6, -33, 162, \ldots$
Incorrect First Option: Since $a_{2}$ does not match the second term of the sequence, which is $-33$ , the first option is incorrect.
Test Second Option: Test the second option. $\newline$ Given: $a_{1}=6$ and $a_{n}=-5a_{n-1}-3$ $\newline$ Calculate $a_{2}$ using the given formula: $a_{2}=-5a_{1}-3=-5\times6-3=-30-3=-33$ $\newline$ Check if this matches the sequence: $6, -33, 162, \ldots$
Verify Second Option: Since $a_{2}$ matches the second term of the sequence, continue to calculate $a_{3}$ to verify the pattern. $\newline$ Calculate $a_{3}$ using the given formula: $a_{3}=-5a_{2}-3=-5*(-33)-3=165-3=162$ $\newline$ Check if this matches the sequence: $6, -33, 162, \ldots$
Correct Second Option: Since $a_{3}$ matches the third term of the sequence, the second option correctly produces the sequence.