Mylo O'Moore

2021-03-08

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 $9=\frac{10}{11}n$

Maciej Morrow

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 $=-\frac{10}{11}$
$\sum =-\frac{7800}{11}$

star233

Result:
$\frac{4950}{11}$
Solution:
$9=\frac{10}{11}n$
We can rewrite the equation as:
$\frac{10}{11}n=9$
To identify whether the sequence is arithmetic, geometric, or neither, we need to analyze the relationship between the terms.
For an arithmetic sequence, the difference between consecutive terms is constant. Let's check if the given sequence follows this pattern:
$\frac{10}{11}\left(n+1\right)-\frac{10}{11}n=\frac{10}{11}n+\frac{10}{11}-\frac{10}{11}n=\frac{10}{11}$
Since the difference between consecutive terms is constant and equal to $\frac{10}{11}$, we can conclude that the given sequence is an arithmetic sequence.
Now, let's find the common difference. In an arithmetic sequence, the common difference is the constant value by which each term is increased or decreased. In this case, the common difference is $\frac{10}{11}$.
Next, we'll find the sum of the first 50 terms of the sequence. The sum of an arithmetic sequence can be calculated using the following formula:
${S}_{n}=\frac{n}{2}\left(2a+\left(n-1\right)d\right)$
Where:
- ${S}_{n}$ is the sum of the first $n$ terms
- $n$ is the number of terms
- $a$ is the first term
- $d$ is the common difference
In our case, the first term $a$ is 9, the number of terms $n$ is 50, and the common difference $d$ is $\frac{10}{11}$. Let's substitute these values into the formula:
${S}_{50}=\frac{50}{2}\left(2·9+\left(50-1\right)·\frac{10}{11}\right)$
Simplifying the expression:
${S}_{50}=25\left(18+49·\frac{10}{11}\right)$
${S}_{50}=25\left(18+\frac{490}{11}\right)$
${S}_{50}=25\left(18+\frac{490}{11}\right)$
${S}_{50}=25·\frac{198}{11}$
${S}_{50}=\frac{4950}{11}$
Therefore, the sum of the first 50 terms of the given arithmetic sequence is $\frac{4950}{11}$.

karton

Step 1:
The given sequence is: $9=\frac{10}{11}n$
To identify the type of sequence, we'll examine the pattern of the terms.
The equation $9=\frac{10}{11}n$ suggests that the terms of the sequence are represented by the variable $n$.
To check if the sequence is arithmetic or geometric, we'll look for a constant difference or ratio between consecutive terms.
For an arithmetic sequence, the difference between consecutive terms is constant. Let's calculate the difference between the terms:
$d=\frac{10}{11}n-\frac{10}{11}\left(n-1\right)$
Simplifying this expression:
$d=\frac{10}{11}n-\frac{10}{11}n+\frac{10}{11}$
$d=\frac{10}{11}$
Since the difference $d$ is constant and equal to $\frac{10}{11}$, the sequence is arithmetic.
Step 2:
Now, let's find the sum of the first 50 terms of the arithmetic sequence.
The formula to calculate the sum of an arithmetic sequence is:
${S}_{n}=\frac{n}{2}\left(2a+\left(n-1\right)d\right)$
where ${S}_{n}$ is the sum of the first $n$ terms, $a$ is the first term, $d$ is the common difference, and $n$ is the number of terms.
Step 3:
In our case, $a=9$ (the first term) and $d=\frac{10}{11}$ (the common difference). We are interested in finding the sum of the first 50 terms ($n=50$).
Plugging in the values:
${S}_{50}=\frac{50}{2}\left(2·9+\left(50-1\right)·\frac{10}{11}\right)$
Simplifying:
${S}_{50}=25\left(18+49·\frac{10}{11}\right)$
${S}_{50}=25\left(18+\frac{490}{11}\right)$
${S}_{50}=25\left(18+\frac{44·10}{11}\right)$
${S}_{50}=25\left(18+\frac{440}{11}\right)$
${S}_{50}=25\left(18+40\right)$
${S}_{50}=25·58$
${S}_{50}=1450$
Therefore, the sum of the first 50 terms of the arithmetic sequence is 1450.
In summary:
- The given sequence $9=\frac{10}{11}n$ is an arithmetic sequence.
- The common difference of the sequence is $\frac{10}{11}$.
- The sum of the first 50 terms of the sequence is 1450.

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