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

Mylo O'Moore

Mylo O'Moore

Answered question

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.
{9=1011n}
What type of sequence is 9=1011n 

Answer & Explanation

Maciej Morrow

Maciej Morrow

Skilled2021-03-09Added 98 answers

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 =1011
Step 2
We will use the sum formula of first n natural numbers.
n=150(91011n)=n=150(91011n)n=150n
=9(50)1011×50(50+1)2
=4501011×50(51)2
=4501275011
=780011
Answer: Aritmetic sequence common difference =1011
=780011

star233

star233

Skilled2023-06-10Added 403 answers

Result:
495011
Solution:
9=1011n
We can rewrite the equation as:
1011n=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:
1011(n+1)1011n=1011n+10111011n=1011
Since the difference between consecutive terms is constant and equal to 1011, 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 1011.
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:
Sn=n2(2a+(n1)d)
Where:
- Sn 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 1011. Let's substitute these values into the formula:
S50=502(2·9+(501)·1011)
Simplifying the expression:
S50=25(18+49·1011)
S50=25(18+49011)
S50=25(18+49011)
S50=25·19811
S50=495011
Therefore, the sum of the first 50 terms of the given arithmetic sequence is 495011.
karton

karton

Expert2023-06-10Added 613 answers

Step 1:
The given sequence is: 9=1011n
To identify the type of sequence, we'll examine the pattern of the terms.
The equation 9=1011n 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=1011n1011(n1)
Simplifying this expression:
d=1011n1011n+1011
d=1011
Since the difference d is constant and equal to 1011, 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:
Sn=n2(2a+(n1)d)
where Sn 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=1011 (the common difference). We are interested in finding the sum of the first 50 terms (n=50).
Plugging in the values:
S50=502(2·9+(501)·1011)
Simplifying:
S50=25(18+49·1011)
S50=25(18+49011)
S50=25(18+44·1011)
S50=25(18+44011)
S50=25(18+40)
S50=25·58
S50=1450
Therefore, the sum of the first 50 terms of the arithmetic sequence is 1450.
In summary:
- The given sequence 9=1011n is an arithmetic sequence.
- The common difference of the sequence is 1011.
- The sum of the first 50 terms of the sequence is 1450.

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