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

Identify Given Sequence: Identify the given sequence and the recursive formula. We are given the first term of the sequence, a_(1)=4, and the recursive formula a_(n)=5a_(n-1)-n, which tells us how to find any term in the sequence based on the previous term.Find Second Term: Find the second term of the sequence using the recursive formula. To find a_(2), we use the formula with n=2: a_(2) = 5a_(2-1) - 2 a_(2) = 5a_(1) - 2 a_(2) = 5*4 - 2 a_(2) = 20 - 2 a_(2) = 18Find Third Term: Find the third term of the sequence using the recursive formula. To find a_(3), we use the formula with n=3: a_(3) = 5a_(3-1) - 3 a_(3) = 5a_(2) - 3 a_(3) = 5*18 - 3 a_(3) = 90 - 3 a_(3) = 87Find Fourth Term: Find the fourth term of the sequence using the recursive formula. To find a_(4), we use the formula with n=4: a_(4) = 5a_(4-1) - 4 a_(4) = 5a_(3) - 4 a_(4) = 5*87 - 4 a_(4) = 435 - 4 a_(4) = 431Find Fifth Term: Find the fifth term of the sequence using the recursive formula. To find a_(5), we use the formula with n=5: a_(5) = 5a_(5-1) - 5 a_(5) = 5a_(4) - 5 a_(5) = 5*431 - 5 a_(5) = 2155 - 5 a_(5) = 2150

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Math Problems

Grade 8

Write and solve direct variation equations

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Q. If

a_{1}=4

and

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

then find the value of

a_{5}

\newline

Answer:

Identify Given Sequence: Identify the given sequence and the recursive formula. $\newline$ We are given the first term of the sequence, $a_{1}=4$ , and the recursive formula $a_{n}=5a_{n-1}-n$ , which tells us how to find any term in the sequence based on the previous term.
Find Second Term: Find the second term of the sequence using the recursive formula. $\newline$ To find $a_{2}$ , we use the formula with $n=2$ : $\newline$ $a_{2} = 5a_{2-1} - 2$ $\newline$ $a_{2} = 5a_{1} - 2$ $\newline$ $a_{2} = 5\times 4 - 2$ $\newline$ $a_{2} = 20 - 2$ $\newline$ $a_{2} = 18$
Find Third Term: Find the third term of the sequence using the recursive formula. $\newline$ To find $a_{3}$ , we use the formula with $n=3$ : $\newline$ $a_{3} = 5a_{3-1} - 3$ $\newline$ $a_{3} = 5a_{2} - 3$ $\newline$ $a_{3} = 5\times 18 - 3$ $\newline$ $a_{3} = 90 - 3$ $\newline$ $a_{3} = 87$
Find Fourth Term: Find the fourth term of the sequence using the recursive formula. $\newline$ To find $a_{4}$ , we use the formula with $n=4$ : $\newline$ $a_{4} = 5a_{4-1} - 4$ $\newline$ $a_{4} = 5a_{3} - 4$ $\newline$ $a_{4} = 5\times87 - 4$ $\newline$ $a_{4} = 435 - 4$ $\newline$ $a_{$4$} = $431$
Find Fifth Term: Find the fifth term of the sequence using the recursive formula.$\newline$To find $a_{5}$, we use the formula with $n=5$:$\newline$$a_{5} = 5a_{5-1} - 5$$\newline$$a_{5} = 5a_{4} - 5$$\newline$$a_{5} = 5\times431 - 5$$\newline$$a_{5} = 2155 - 5$$\newline$$a_{5} = 2150$