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Write an expression to describe the sequence below. Use $n$ to represent the position of a term in the sequence, where $n = 1$ for the first term. $\newline$ $–83, –166, –249, –332, \dots$ $\newline$ $a_n =$ _____

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Write variable expressions for arithmetic sequences

Full solution

Q. Write an expression to describe the sequence below. Use

n

to represent the position of a term in the sequence, where

n = 1

for the first term.

\newline

–83, –166, –249, –332, \dots

\newline

a_n =

_____

Identify type of sequence: Identify the type of sequence. $\newline$ We have: $\newline$ $-83, -166, -249, -332, \ldots$ $\newline$ Is the given sequence geometric or arithmetic? $\newline$ To determine this, we look at the differences between consecutive terms. $\newline$ $-166 - (-83) = -83$ $\newline$ $-249 - (-166) = -83$ $\newline$ $-332 - (-249) = -83$ $\newline$ Since there is a common difference between consecutive terms, the given sequence is arithmetic.
Determine values of $a_1$ and $d$ : Determine the values of $a_1$ and $d$ of the sequence. $\newline$ The first term, $a_1 = -83$ $\newline$ Common difference, $d = -166 - (-83) = -83$
Write nth term formula: Write the formula for the nth term of an arithmetic sequence. $\newline$ The formula for the nth term of an arithmetic sequence is: $\newline$ $a_n = a_1 + (n-1)d$
Substitute values into formula: Substitute the values of $a_1$ and $d$ into the formula. $a_1 = -83$ $d = -83$ $a_n = a_1 + (n-1)d$ $a_n = -83 + (n-1)(-83)$
Simplify expression: Simplify the expression. $\newline$ $a_n = -83 - 83(n-1)$ $\newline$ $a_n = -83 - 83n + 83$ $\newline$ $a_n = -83n$ $\newline$ Since $-83 + 83$ equals $0$ , the simplified expression is: $\newline$ $a_n = -83n$

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