If a_(1)=4,a_(2)=1 and a_(n)=2a_(n-1)+3a_(n-2) then find the value of a_(4). Answer:

Understand Formula and Conditions: Understand the recursive formula and initial conditions. The recursive formula given is a_(n)=2a_(n-1)+3a_(n-2). This means that each term in the sequence is generated by multiplying the previous term by 2 and the term before that by 3, then adding the results. We are given a_(1)=4 and a_(2)=1 as initial conditions.Find Third Term: Find the third term of the sequence using the recursive formula. To find a_(3), we use the formula with n=3, which means we need a_(2) and a_(1). a_(3) = 2a_(2) + 3a_(1) a_(3) = 2(1) + 3(4) a_(3) = 2 + 12 a_(3) = 14Find Fourth Term: Find the fourth term of the sequence using the recursive formula. Now that we have a_(3), we can find a_(4) using the formula with n=4, which means we need a_(3) and a_(2). a_(4) = 2a_(3) + 3a_(2) a_(4) = 2(14) + 3(1) a_(4) = 28 + 3 a_(4) = 31

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Grade 8

Write and solve direct variation equations

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Q. If

a_{1}=4, a_{2}=1

and

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

then find the value of

a_{4}

\newline

Answer:

Understand Formula and Conditions: Understand the recursive formula and initial conditions. $\newline$ The recursive formula given is $a_{n}=2a_{n-1}+3a_{n-2}$ . This means that each term in the sequence is generated by multiplying the previous term by $2$ and the term before that by $3$ , then adding the results. We are given $a_{1}=4$ and $a_{2}=1$ as initial conditions.
Find Third Term: Find the third term of the sequence using the recursive formula. $\newline$ To find $a_{3}$ , we use the formula with $n=3$ , which means we need $a_{2}$ and $a_{1}$ . $\newline$ $a_{3} = 2a_{2} + 3a_{1}$ $\newline$ $a_{3} = 2(1) + 3(4)$ $\newline$ $a_{3} = 2 + 12$ $\newline$ $a_{3} = 14$
Find Fourth Term: Find the fourth term of the sequence using the recursive formula. $\newline$ Now that we have $a_{3}$ , we can find $a_{4}$ using the formula with $n=4$ , which means we need $a_{3}$ and $a_{2}$ . $\newline$ $a_{4} = 2a_{3} + 3a_{2}$ $\newline$ $a_{4} = 2(14) + 3(1)$ $\newline$ $a_{4} = 28 + 3$ $\newline$ $a_{4} = 31$