For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence). $\newline$ $7,-35,175, \ldots$ $\newline$ $-5$ $\newline$ $-42$ $\newline$ $-\frac{1}{5}$ $\newline$ $42$

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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

7,-35,175, \ldots

\newline

-5

\newline

-42

\newline

-\frac{1}{5}

\newline

42

Check Common Difference: To determine whether the sequence is arithmetic or geometric, we need to examine the relationship between consecutive terms. $\newline$ Let's first check if there is a common difference by subtracting the second term from the first term and the third term from the second term. $\newline$ Common difference $d$ between the first and second term: $d = -35 - 7$
Calculate Common Difference: Perform the subtraction to find the common difference between the first and second term. $\newline$ $d = -35 - 7 = -42$
Check Common Ratio: Now, let's find the common difference between the second and third term. $\newline$ $d = 175 - (-35)$
Calculate Common Ratio: Perform the subtraction to find the common difference between the second and third term. $\newline$ $d = 175 - (-35) = 175 + 35 = 210$
Sequence Type Determination: Since the common difference between the first and second term is not the same as the common difference between the second and third term, the sequence is not arithmetic. $\newline$ Now, let's check if there is a common ratio by dividing the second term by the first term and the third term by the second term. $\newline$ Common ratio $r$ between the first and second term: $r = -\frac{35}{7}$
Sequence Type Determination: Since the common difference between the first and second term is not the same as the common difference between the second and third term, the sequence is not arithmetic. $\newline$ Now, let's check if there is a common ratio by dividing the second term by the first term and the third term by the second term. $\newline$ Common ratio $r$ between the first and second term: $r = -35 / 7$ Perform the division to find the common ratio between the first and second term. $\newline$ $r = -35 / 7 = -5$
Sequence Type Determination: Since the common difference between the first and second term is not the same as the common difference between the second and third term, the sequence is not arithmetic. $\newline$ Now, let's check if there is a common ratio by dividing the second term by the first term and the third term by the second term. $\newline$ Common ratio $r$ between the first and second term: $r = -35 / 7$ Perform the division to find the common ratio between the first and second term. $\newline$ $r = -35 / 7 = -5$ Now, let's find the common ratio between the second and third term. $\newline$ $r = 175 / (-35)$
Sequence Type Determination: Since the common difference between the first and second term is not the same as the common difference between the second and third term, the sequence is not arithmetic. $\newline$ Now, let's check if there is a common ratio by dividing the second term by the first term and the third term by the second term. $\newline$ Common ratio $r$ between the first and second term: $r = -35 / 7$ Perform the division to find the common ratio between the first and second term. $\newline$ $r = -35 / 7 = -5$ Now, let's find the common ratio between the second and third term. $\newline$ $r = 175 / (-35)$ Perform the division to find the common ratio between the second and third term. $\newline$ $r = 175 / (-35) = -5$
Sequence Type Determination: Since the common difference between the first and second term is not the same as the common difference between the second and third term, the sequence is not arithmetic. $\newline$ Now, let's check if there is a common ratio by dividing the second term by the first term and the third term by the second term. $\newline$ Common ratio $r$ between the first and second term: $r = -35 / 7$ Perform the division to find the common ratio between the first and second term. $\newline$ $r = -35 / 7 = -5$ Now, let's find the common ratio between the second and third term. $\newline$ $r = 175 / (-35)$ Perform the division to find the common ratio between the second and third term. $\newline$ $r = 175 / (-35) = -5$ Since the common ratio between the first and second term is the same as the common ratio between the second and third term, the sequence is geometric with a common ratio of $-5$ .