# Determine whether the given sequence is arithmetic, geometric, or neither.

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference, if it is geometric, find the common ratio. If the sequence is arithmetic or geometric,
find the sum of the first 50 terms.
$\left\{9=\frac{10}{11}n\right\}$
What type of sequence is $9=\frac{10}{11}n$

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Step 1
Since the expression is a linear function of "n". So it is arithmetic sequence.
Common difference is the coefficient of "n" that is $=-\frac{10}{11}$
Step 2
We will use the sum formula of first n natural numbers.
$\sum _{n=1}^{50}\left(9-\frac{10}{11}n\right)=\sum _{n=1}^{50}\left(9-\frac{10}{11n}\right)\sum _{n=1}^{50}n$
$=9\left(50\right)-\frac{10}{11}×\frac{50\left(50+1\right)}{2}$
$=450-\frac{10}{11}×\frac{50\left(51\right)}{2}$
$=450-\frac{12750}{11}$
$=-\frac{7800}{11}$
Answer: Aritmetic sequence common difference $=-\frac{10}{11}$
$\sum =-\frac{7800}{11}$