If a_(1)=5,a_(2)=1 and a_(n)=3a_(n-1)-2a_(n-2) then find the value of a_(6). Answer:

Understand Formula and Conditions: Understand the recursive formula and initial conditions. The recursive formula given is a_(n)=3a_(n-1)-2a_(n-2), which means each term is calculated by tripling the previous term and subtracting twice the term before the previous one. We are given a_(1)=5 and a_(2)=1 as initial conditions.Calculate Third Term: Calculate the third term using the recursive formula. a_(3) = 3a_(2) - 2a_(1) a_(3) = 3(1) - 2(5) a_(3) = 3 - 10 a_(3) = -7Calculate Fourth Term: Calculate the fourth term using the recursive formula. a_(4) = 3a_(3) - 2a_(2) a_(4) = 3(-7) - 2(1) a_(4) = -21 - 2 a_(4) = -23Calculate Fifth Term: Calculate the fifth term using the recursive formula. a_(5) = 3a_(4) - 2a_(3) a_(5) = 3(-23) - 2(-7) a_(5) = -69 + 14 a_(5) = -55Calculate Sixth Term: Calculate the sixth term using the recursive formula. a_(6) = 3a_(5) - 2a_(4) a_(6) = 3(-55) - 2(-23) a_(6) = -165 + 46 a_(6) = -119

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Math Problems

Grade 8

Write and solve direct variation equations

Full solution

Q. If

a_{1}=5, a_{2}=1

and

a_{n}=3 a_{n-1}-2 a_{n-2}

then find the value of

a_{6}

\newline

Answer:

Understand Formula and Conditions: Understand the recursive formula and initial conditions. $\newline$ The recursive formula given is $a_{n}=3a_{n-1}-2a_{n-2}$ , which means each term is calculated by tripling the previous term and subtracting twice the term before the previous one. We are given $a_{1}=5$ and $a_{2}=1$ as initial conditions.
Calculate Third Term: Calculate the third term using the recursive formula. $\newline$ $a_{3} = 3a_{2} - 2a_{1}$ $\newline$ $a_{3} = 3(1) - 2(5)$ $\newline$ $a_{3} = 3 - 10$ $\newline$ $a_{3} = -7$
Calculate Fourth Term: Calculate the fourth term using the recursive formula. $\newline$ $a_{4} = 3a_{3} - 2a_{2}$ $\newline$ $a_{4} = 3(-7) - 2(1)$ $\newline$ $a_{4} = -21 - 2$ $\newline$ $a_{4} = -23$
Calculate Fifth Term: Calculate the fifth term using the recursive formula. $\newline$ $a_{5} = 3a_{4} - 2a_{3}$ $\newline$ $a_{5} = 3(-23) - 2(-7)$ $\newline$ $a_{5} = -69 + 14$ $\newline$ $a_{5} = -55$
Calculate Sixth Term: Calculate the sixth term using the recursive formula. $\newline$ $a_{6} = 3a_{5} - 2a_{4}$ $\newline$ $a_{6} = 3(-55) - 2(-23)$ $\newline$ $a_{6} = -165 + 46$ $\newline$ $a_{6} = -119$