The sum of the first n term of ab arithmetic series is given by Sn=5/2n^2+7/2n

Determine the first term and the common different and the tenth term

Sn = 5/2 n^2 + 7/2 n = n/2 (5n+7)

Sn = n/2 (2a+(n-1)d)

5(n-1)+5+7 = 5(n-1)+12
a=6 d=5

T10 = 12+9*5 = 57

check:
6 = 1/2 (12+0)
6+11 = 17 = 2/2 (12+5)
6+11+16 = 33 = 3/2 (12+10)
looking good.

To determine the first term and the common difference of an arithmetic series, as well as the tenth term, we need to use the given formula for the sum of the first n terms of an arithmetic series, which is given by:

Sn = (n/2)(2a + (n-1)d)

In this formula, Sn represents the sum of the first n terms, a represents the first term, and d represents the common difference.

Comparing this formula to the given formula, we can see that:

Sn = 5/2n^2 + 7/2n
2a + (n-1)d = 5/2n^2 + 7/2n

To determine the first term (a) and common difference (d), we can create a system of equations using the above expressions:

Equation 1: 2a + (n-1)d = 5/2n^2 + 7/2n
Equation 2: Sn = 5/2n^2 + 7/2n

Since we want to find the first term and common difference, let's solve the system of equations for a and d.

To find the first term (a), substitute Sn in Equation 1 with the given formula:

2a + (n-1)d = Sn
2a + (n-1)d = 5/2n^2 + 7/2n

To find the common difference (d), we can rearrange Equation 1 as follows:

d = (Sn - 2a) / (n-1)

Now, let's find the first term and common difference using the above equations.

To find the first term (a), we need to know the value of Sn, n, and substitute them into the equation:

Sn = 5/2n^2 + 7/2n
n = 10 (since we want to find the tenth term)
Sn = 5/2(10^2) + 7/2(10)
Sn = 250 + 35
Sn = 285

Substituting n and Sn into Equation 1:

2a + 9d = 285

To find the common difference (d), we substitute the values of Sn, a, and n into Equation 2:

d = (285 - 2a) / (10-1)

Now, you can solve the above two equations to find the values of a and d using any method of solving simultaneous equations such as substitution or elimination.

Once you have the values of a and d, you can find the tenth term by substituting n = 10 into the arithmetic sequence formula:

Tn = a + (n-1)d

Substitute the values of a, d, and n:

T10 = a + (10-1)d

Calculate the value of T10 to find the tenth term.