If a_(1)=1 and a_(n)=na_(n-1)+4 then find the value of a_(5). Answer:

Understand formula and condition: Understand the recursive formula and the initial condition. The recursive formula given is a_(n)=na_(n-1)+4, and the initial condition is a_(1)=1. This means that each term in the sequence is n times the previous term plus 4.Find a_(2): Find the value of a_(2) using the recursive formula. Since a_(1)=1, we can find a_(2) by plugging n=2 into the formula: a_(2) = 2 * a_(1) + 4 a_(2) = 2 * 1 + 4 a_(2) = 2 + 4 a_(2) = 6Find a_(3): Find the value of a_(3) using the recursive formula. Now that we have a_(2), we can find a_(3) by plugging n=3 into the formula: a_(3) = 3 * a_(2) + 4 a_(3) = 3 * 6 + 4 a_(3) = 18 + 4 a_(3) = 22Find a_(4): Find the value of a_(4) using the recursive formula. With a_(3) found, we can find a_(4) by plugging n=4 into the formula: a_(4) = 4 * a_(3) + 4 a_(4) = 4 * 22 + 4 a_(4) = 88 + 4 a_(4) = 92Find a_(5): Find the value of a_(5) using the recursive formula. Finally, we can find a_(5) by plugging n=5 into the formula: a_(5) = 5 * a_(4) + 4 a_(5) = 5 * 92 + 4 a_(5) = 460 + 4 a_(5) = 464

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Math Problems

Grade 8

Write and solve direct variation equations

Full solution

Q. If

a_{1}=1

and

a_{n}=n a_{n-1}+4

then find the value of

a_{5}

\newline

Answer:

Understand formula and condition: Understand the recursive formula and the initial condition. $\newline$ The recursive formula given is $a_{n}=na_{n-1}+4$ , and the initial condition is $a_{1}=1$ . This means that each term in the sequence is $n$ times the previous term plus $4$ .
Find $a_{2}$ : Find the value of $a_{2}$ using the recursive formula. $\newline$ Since $a_{1}=1$ , we can find $a_{2}$ by plugging $n=2$ into the formula: $\newline$ $a_{2} = 2 \times a_{1} + 4$ $\newline$ $a_{2} = 2 \times 1 + 4$ $\newline$ $a_{2} = 2 + 4$ $\newline$ $a_{2} = 6$
Find $a_{3}$ : Find the value of $a_{3}$ using the recursive formula. $\newline$ Now that we have $a_{2}$ , we can find $a_{3}$ by plugging $n=3$ into the formula: $\newline$ $a_{3} = 3 \times a_{2} + 4$ $\newline$ $a_{3} = 3 \times 6 + 4$ $\newline$ $a_{3} = 18 + 4$ $\newline$ $a_{3} = 22$
Find $a_{4}$ : Find the value of $a_{4}$ using the recursive formula. $\newline$ With $a_{3}$ found, we can find $a_{4}$ by plugging $n=4$ into the formula: $\newline$ $a_{4} = 4 \times a_{3} + 4$ $\newline$ $a_{4} = 4 \times 22 + 4$ $\newline$ $a_{4} = 88 + 4$ $\newline$ $a_{4} = 92$
Find $a_{5}$ : Find the value of $a_{5}$ using the recursive formula. $\newline$ Finally, we can find $a_{5}$ by plugging $n=5$ into the formula: $\newline$ $a_{5} = 5 \times a_{4} + 4$ $\newline$ $a_{5} = 5 \times 92 + 4$ $\newline$ $a_{5} = 460 + 4$ $\newline$ $a_{5} = 464$