The sum of n terms of an a.p. Is 3n2+2n,then find its p'th term.
n/2 (2a + (n-1)d) = 3n^2+2n
2an + dn^2 - dn = 6n^2+4n
dn^2 + (2a-d)n = 6n^2+4n
d = 6
2a-d = 4, so a = 5
Tp = 5+(p-1)(6) = 6p-1
check: the AP is
5,11,17,23,29,...
S1 = 5 = 3*1^2+2*1
S2 = 16 = 3*2^2+2*2
S3 = 33 = 3*3^2+2*3
S4 = 56 = 3*4^2+2*4
...
To find the pth term of an arithmetic progression (AP) when the sum of n terms is given, we need to follow these steps:
Step 1: Determine the formula for the sum of the first n terms of an AP.
The formula for the sum of the first n terms of an AP is given by: Sn = (n/2) * (2 * a + (n - 1) * d)
Where:
Sn = Sum of the first n terms
n = Number of terms
a = First term of the AP
d = Common difference between the terms
Step 2: Compare the given sum of n terms with the formula to find the values of a and d.
Given:
Sum of n terms = 3n^2 + 2n
Comparing this with the formula, we can see that:
3n^2 + 2n = (n/2) * (2 * a + (n - 1) * d)
Comparing coefficients of corresponding terms, we get:
3 = d/2
2 = a + (n - 1) * d
Step 3: Solve the equations to find the values of a and d.
From equation 1, we can solve for d:
3 = d/2
d = 6
Substituting the value of d in equation 2, we can solve for a:
2 = a + (n - 1) * 6
2 = a + 6n - 6
a + 6n = 8
Step 4: Use the values of a and d to find the pth term.
The pth term of an AP is given by: Tp = a + (p - 1) * d
Substituting the values we found, we get:
Tp = 8 + (p - 1) * 6
Tp = 8 + 6p - 6
Tp = 6p + 2
Therefore, the pth term of the arithmetic progression is 6p + 2.