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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. $\newline$ $77, 154, 231, 308, \ldots$ $\newline$ $a_n =$ _____

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Write variable expressions for arithmetic sequences

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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

77, 154, 231, 308, \ldots

\newline

a_n =

_____

Sequence Type: We have: $\newline$ $77, 154, 231, 308, \ldots$ $\newline$ Is the given sequence geometric or arithmetic? $\newline$ To determine this, we can look at the differences or ratios between terms.
Calculate Differences: $77, 154, 231, 308, \ldots$ $\newline$ Calculate the difference between consecutive terms to see if it is constant. $\newline$ Difference between second and first term: $154 - 77 = 77$ $\newline$ Difference between third and second term: $231 - 154 = 77$ $\newline$ Difference between fourth and third term: $308 - 231 = 77$ $\newline$ Since the differences are constant, the sequence is arithmetic.
Find $a_1$ and $d$ : Determine the values of $a_1$ and $d$ of the sequence. $\newline$ The first term, $a_1 = 77$ $\newline$ Common difference, $d = 77$ (as calculated in the previous step)
Arithmetic Sequence Formula: We have the formula for the $n$ th term of an arithmetic sequence: $a_n = a_1 + (n-1)d$ $a_1 = 77$ $d = 77$ Write an expression to describe $77, 154, 231, 308, \ldots$ $a_n = 77 + (n-1)\times77$
Simplify Expression: Simplify the expression: $\newline$ $a_{n} = 77 + 77n - 77$ $\newline$ $a_{n} = 77n$ $\newline$ This is the expression that describes the sequence.

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