For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence). $\newline$ $x-3, \quad 2 x-3, \quad 3 x-3, \quad \text {.. }$ $\newline$ $-x$ $\newline$ $-1$ $\newline$ $1$ $\newline$ $x$

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Identify arithmetic and geometric series

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Q. For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence).

\newline

x-3, \quad 2 x-3, \quad 3 x-3, \quad \text {.. }

\newline

-x

\newline

-1

\newline

1

\newline

x

Identify Sequence Type: To determine if the sequence is arithmetic or geometric, we need to look at the pattern of the terms given. An arithmetic sequence has a constant difference between terms, while a geometric sequence has a constant ratio between terms. Let's first check for an arithmetic sequence by finding the difference between consecutive terms.
Calculate Difference for $1$ st & $2$ nd Terms: The first term is $x - 3$ . The second term is $2x - 3$ . To find the difference between the second and the first term, we calculate $(2x - 3) - (x - 3)$ .
Calculate Difference for $2$ nd & $3$ rd Terms: Simplifying the difference, we get $(2x - 3) - (x - 3) = 2x - 3 - x + 3 = x$ . The difference between the first two terms is $x$ .
Confirm Arithmetic Sequence: Now let's find the difference between the third and the second term. The third term is $3x - 3$ . We calculate $(3x - 3) - (2x - 3)$ .
Check Last Term Difference: Simplifying this difference, we get $(3x - 3) - (2x - 3) = 3x - 3 - 2x + 3 = x$ . The difference between the second and third terms is also $x$ .
Check Last Term Difference: Simplifying this difference, we get $(3x - 3) - (2x - 3) = 3x - 3 - 2x + 3 = x$ . The difference between the second and third terms is also $x$ . Since the differences between consecutive terms are the same ( $x$ ), we can conclude that the sequence is an arithmetic sequence with a common difference of $x$ .
Check Last Term Difference: Simplifying this difference, we get $(3x - 3) - (2x - 3) = 3x - 3 - 2x + 3 = x$ . The difference between the second and third terms is also $x$ . Since the differences between consecutive terms are the same ( $x$ ), we can conclude that the sequence is an arithmetic sequence with a common difference of $x$ . To confirm our conclusion, let's check the difference between the last given term, $-x$ , and the term before it, $3x - 3$ . We calculate $(-x) - (3x - 3)$ .
Check Last Term Difference: Simplifying this difference, we get $(3x - 3) - (2x - 3) = 3x - 3 - 2x + 3 = x$ . The difference between the second and third terms is also $x$ . Since the differences between consecutive terms are the same ( $x$ ), we can conclude that the sequence is an arithmetic sequence with a common difference of $x$ . To confirm our conclusion, let's check the difference between the last given term, $-x$ , and the term before it, $3x - 3$ . We calculate $(-x) - (3x - 3)$ . Simplifying this difference, we get $(-x) - (3x - 3) = -x - 3x + 3 = -4x + 3$ .