Question
Consider the arithmetic sequence 2x+1;4x+1;6x+1;...
Determine the general term of this sequence in terms of x.
Which term in the sequence is equal to 150x+17
If the sum of the first 21terms of the sequence is equal to 1407, determine the value of x
Determine the general term of this sequence in terms of x.
Which term in the sequence is equal to 150x+17
If the sum of the first 21terms of the sequence is equal to 1407, determine the value of x
Answers
GPT 3.5
To find the general term of the arithmetic sequence, we notice that each term is obtained by adding 2x to the previous term. So, we have the common difference d = 2x.
The first term is given by the expression 2x + 1.
To find which term in the sequence is equal to 150x + 17, we need to find the value of n such that the nth term of the sequence is equal to 150x + 17.
Using the formula for the nth term of an arithmetic sequence, we have:
a_n = a_1 + (n-1)d
where a_n is the nth term, a_1 is the first term, and d is the common difference.
We know that a_n = 150x + 17, a_1 = 2x + 1, and d = 2x.
So we can write the equation:
150x + 17 = 2x + 1 + (n-1)(2x)
Simplifying, we have:
150x + 17 = 2x + 1 + 2nx - 2x
150x + 17 = 1 + 2nx
2nx = 150x + 16
nx = 75x + 8
n = (75x + 8) / x
n = 75 + 8/x
Therefore, the term in the sequence that is equal to 150x + 17 is the (75 + 8/x)th term.
To find the value of x when the sum of the first 21 terms of the sequence is equal to 1407, we use the formula for the sum of an arithmetic series:
S_n = (n/2)(a_1 + a_n)
where S_n is the sum of the first n terms, a_1 is the first term, and a_n is the nth term.
We know that S_n = 1407, n = 21, a_1 = 2x + 1, and a_n = 2x + 1 + (n-1)d.
Substituting the values into the formula, we have:
1407 = (21/2)(2x + 1 + (21-1)(2x))
Simplifying, we have:
1407 = 10.5(2x + 1 + 20(2x))
1407 = 10.5(2x + 1 + 40x)
1407 = 10.5(42x + 1)
1407 = 441x + 10.5
441x = 1396.5
x = 1396.5/441
x = 3.1646
Therefore, the value of x when the sum of the first 21 terms of the sequence is equal to 1407 is approximately 3.1646.
The first term is given by the expression 2x + 1.
To find which term in the sequence is equal to 150x + 17, we need to find the value of n such that the nth term of the sequence is equal to 150x + 17.
Using the formula for the nth term of an arithmetic sequence, we have:
a_n = a_1 + (n-1)d
where a_n is the nth term, a_1 is the first term, and d is the common difference.
We know that a_n = 150x + 17, a_1 = 2x + 1, and d = 2x.
So we can write the equation:
150x + 17 = 2x + 1 + (n-1)(2x)
Simplifying, we have:
150x + 17 = 2x + 1 + 2nx - 2x
150x + 17 = 1 + 2nx
2nx = 150x + 16
nx = 75x + 8
n = (75x + 8) / x
n = 75 + 8/x
Therefore, the term in the sequence that is equal to 150x + 17 is the (75 + 8/x)th term.
To find the value of x when the sum of the first 21 terms of the sequence is equal to 1407, we use the formula for the sum of an arithmetic series:
S_n = (n/2)(a_1 + a_n)
where S_n is the sum of the first n terms, a_1 is the first term, and a_n is the nth term.
We know that S_n = 1407, n = 21, a_1 = 2x + 1, and a_n = 2x + 1 + (n-1)d.
Substituting the values into the formula, we have:
1407 = (21/2)(2x + 1 + (21-1)(2x))
Simplifying, we have:
1407 = 10.5(2x + 1 + 20(2x))
1407 = 10.5(2x + 1 + 40x)
1407 = 10.5(42x + 1)
1407 = 441x + 10.5
441x = 1396.5
x = 1396.5/441
x = 3.1646
Therefore, the value of x when the sum of the first 21 terms of the sequence is equal to 1407 is approximately 3.1646.
Related Questions
The first term of an arithmetic sequence is -1 and the 7^th term is 35 determine
1.The common diff...
reate your own arithmetic sequence. List at least six terms of your sequence. Explain what makes it...
Question 1
Create your own arithmetic sequence. List at least six terms of your sequence. Explain...
Create your own arithmetic sequence. List at least six terms of your sequence. Explain what makes it...