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
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.