the sum of the first nth term of a linear sequence (a-p)is 2n^2-n-1.find the nth term
n/2 (a1+an) = 2n^2-n-1
an = 2/n (2n^2-n-1) - a1
To find the nth term of a linear sequence, we need to determine the pattern or rule that describes how each term is related to the term number.
Given that the sum of the first n terms of the sequence (a-p) is 2n^2 - n - 1, we can use this formula to establish a relationship between the sequence and the term number.
The sum formula for an arithmetic series is:
Sn = (n/2)(a1 + an)
Where Sn represents the sum of the first n terms, a1 is the first term, and an is the nth term.
In this case, we have the sum formula:
2n^2 - n - 1 = (n/2)(a1 + an)
Simplifying the equation, we get:
4n^2 - 2n - 2 = n(a1 + an)
Since the sequence is linear (a-p), an = a1 + (n-1)d, where d is the common difference between consecutive terms in the sequence.
Substituting this into our equation, we have:
4n^2 - 2n - 2 = n(a1 + a1 + (n-1)d)
Expanding and rearranging the equation, we obtain:
4n^2 - 2n - 2 = 2na1 + (n^2 - n)d
Since the expression on the right side of the equation only contains a1 and d, we can equate the coefficients of n^2, n, and the constant term on both sides.
Coefficients of n^2:
4 = d
Coefficients of n:
-2 = a1 + d
Constant terms:
-2 = 2na1
From the constant term equation, we can solve for a1:
-2 = 2na1
a1 = -1/n
Then, substituting a1 into the coefficient of n equation:
-2 = -1/n + d
d = -2 + 1/n
Now that we have determined the values of a1 and d, we can express the nth term (an) as:
an = a1 + (n-1)d
an = -1/n + (n-1)(-2 + 1/n)
an = -1/n - 2n + 2 + 1
an = 3 - (2n + 1)/n
Therefore, the nth term of the given linear sequence is 3 - (2n + 1)/n.