the sum of n terms of an a.p. is 3n^2+2n, then find its pth term.
(answer 6p-1)
To find the pth term of the arithmetic progression (A.P), we need to find the common difference (d) first.
Given that the sum of n terms of the A.P is 3n^2 + 2n, we can write the formula for the sum of n terms of an A.P as:
Sn = (n/2) * [2a + (n - 1)d], where Sn is the sum of n terms, a is the first term, n is the number of terms, and d is the common difference.
Comparing this formula with the given expression, we can equate them:
3n^2 + 2n = (n/2) * [2a + (n - 1)d]
Simplifying the equation, we get:
3n^2 + 2n = n*(a + a + (n - 1)d)
3n^2 + 2n = n(2a + (n - 1)d)
Now, we need to identify the coefficients of n^2 and n, and equate them to find the value of d.
Coefficient of n^2:
3 = 2a
Coefficient of n:
2 = (n - 1)d
From the first equation, we can simplify it to find the value of a:
a = 3/2
Now, substituting the value of a into the second equation, we can find d:
2 = (n - 1)d
2 = (n - 1) * (3/2)
2 = (3n - 3)/2
4 = (3n - 3)
3n = 7
n = 7/3
Therefore, the common difference (d) of the given A.P is 7/3.
Now that we have the values of a (the first term) and d (the common difference), we can find the pth term using the formula for the nth term of an A.P:
An = a + (n - 1)d
Substituting the values of a and d, we get:
Ap = (3/2) + ((p - 1) * (7/3))
Ap = (3/2) + (7p - 7)/3
To simplify the expression, we can multiply all terms by 6 to eliminate the fractions:
6Ap = 9 + 14p - 14
6Ap = 14p - 5
Finally, rearranging the equation to isolate Ap, we get the desired result:
6Ap = 14p - 5
Ap = (14p - 5)/6
Ap = 6p - 5/6
Ap = 6p - 1
Hence, the pth term of the given A.P is 6p - 1.