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

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:

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Write and solve direct variation equations

Full solution

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 $a_{2}$ . $\newline$ $a_{4} = 2a_{3} + 3a_{2}$ $\newline$ $a_{4} = 2(14) + 3(1)$ $\newline$ $a_{4} = 28 + 3$ $\newline$ $a_{4} = 31$

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