The 10th term of an arithmetic series is 34, and the sum of the first 20 terms is 710. Determine the 25th term.

Use your formulas

a+9d = 34 (#1)

(20/2)[2a + 19d] = 710
2a + 19d = 71 (#2)

solve the two equations.
I would use #1 as a = 34-9d and sub into #2
Let me know what you got.

The sum of members of a arithmetic progression is:

Sn=(n/2)*[2a1+(n-1)*d]

a1-first number in arithmetic progression

d-common difference of successive members

n-numbers of members

In this case:
n=20 , (n/2)=10 , n-1=19

a1=a10-9*d
a1=34-9*d

Sn=(n/2)*[2a1+(n-1)*d]
=10*[2*(34-9*d)+19*d]
=10*(68-18d+19d)=10*(68+d)=680+10d
Sn=S20
S20=710
710=680+10d
10d=710-680=30
10d=30 Divided with 10
d=30/10

d=3

a1=a10-9d
a1=34-9*3
a1=34-27

a1=7

a25=a1+(25-1)*d
a25=7+24*3
a25=7+72

a25=79

So members of that Arithmetic progression is:

7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,79

For more information about Arithmetic progression go to wikipedia and type"Arithmetic progression"

a1=34-9*d

Becouse nth member in arithmetic progression is:
an=a1+(n-1)*d

a10=a1+(10-1)*d
a10=a1+9*d
a10-9*d=a1
a
a1=a10-9*d
a1=34-9*d

an=a1+(n-1)*d
a25=a1+(25-1)*d
a25=a1+24*d
a25=7+24*3
a25=7+72=79

To determine the 25th term of the arithmetic series, we need to find the common difference (d) and use it to calculate the 25th term.

First, let's find the common difference using the given information. We know that the 10th term is 34. The formula for the nth term of an arithmetic series is:

an = a1 + (n - 1)d

where an is the nth term, a1 is the first term, n is the position of the term, and d is the common difference.

Substituting the given values, we get:

34 = a1 + (10 - 1)d

Simplifying the equation, we have:

34 = a1 + 9d ------(1)

Next, let's find the sum of the first 20 terms, which is 710. The sum of an arithmetic series is given by the formula:

Sn = (n/2)(a1 + an)

where Sn is the sum of the first n terms.

Substituting the given values, we get:

710 = (20/2)(a1 + a25) ------(2)

Now, we have two equations with two variables (a1 and d) that we can solve simultaneously to find their values.

From equation (1), we have:

a1 = 34 - 9d

Substituting this value of a1 into equation (2), we get:

710 = (20/2)(34 - 9d + a25)

Simplifying the equation, we have:

710 = 10(34 - 9d + a25)

Distributing, we get:

710 = 340 - 90d + 10a25

Rearranging the terms, we have:

90d - 10a25 = -340

Dividing both sides of the equation by 10, we get:

9d - a25 = -34 ------(3)

Now, we have a system of equations (equations (1) and (3)) that we can solve simultaneously.

Subtracting equation (3) from equation (1), we get:

34 - (-34) = a1 + 9d - (9d - a25)

Simplifying the equation, we have:

68 = 2a1 + a25

Now, we have one equation (equation (4)) with two variables (a1 and a25) that we can use to find the value of the 25th term.

Since we need to determine the 25th term, we can isolate a25 by solving equation (4) for a25:

a25 = 68 - 2a1

Now, we can substitute this value of a25 back into equation (3) to solve for the common difference (d):

9d - (68 - 2a1) = -34

Simplifying the equation, we have:

9d - 68 + 2a1 = -34

Combining like terms, we get:

9d + 2a1 = 34 + 68

Simplifying further, we have:

9d + 2a1 = 102 ------(5)

Now, we have two equations (equations (4) and (5)) with two variables (a1 and d) in a system that we can solve simultaneously.

To solve equations (4) and (5) simultaneously, we can substitute the value of a1 from equation (4) into equation (5):

9d + 2(68 - 2a1) = 102

Simplifying the equation, we have:

9d + 136 - 4a1 = 102

Combining like terms, we get:

9d - 4a1 = 102 - 136

Simplifying further, we have:

9d - 4a1 = -34 ------(6)

Now, we have one equation (equation (6)) with one variable (d) that we can solve to find its value.

To solve equation (6) for d, we can substitute the value of a1 from equation (4) into equation (6):

9d - 4(68 - 2a1) = -34

Simplifying the equation, we have:

9d - 4(68 - 2(68 - 2a1)) = -34

Simplifying further, we have:

9d - 272 + 8(68 - 2a1) = -34

Expanding the expression, we get:

9d - 272 + 8(68) - 8(2a1) = -34

Simplifying, we have:

9d - 272 + 544 - 16a1 = -34

Combining like terms, we get:

9d - 16a1 + 272 = -34 + 544

Simplifying further, we have:

9d - 16a1 + 272 = 510

Rearranging the terms, we have:

9d - 16a1 = 510 - 272

Simplifying, we have:

9d - 16a1 = 238 ------(7)

Now, we have one equation (equation (7)) with one variable (d) that we can solve to find its value.

To solve equation (7) for d, we can isolate d by solving equation (7) for a1 and substituting it back into equation (5):

From equation (7), we have:

9d = 16a1 + 238

Dividing both sides of the equation by 9, we get:

d = (16a1 + 238)/9

Now, we can substitute this value of d back into equation (5) to solve for a1:

9((16a1 + 238)/9) + 2a1 = 102

Simplifying the equation, we have:

16a1 + 238 + 2a1 = 102

Combining like terms, we get:

18a1 + 238 = 102

Subtracting 238 from both sides of the equation, we get:

18a1 = 102 - 238

Simplifying, we have:

18a1 = -136

Dividing both sides of the equation by 18, we get:

a1 = -136/18

Simplifying, we have:

a1 = -68/9

Now that we have the value of a1, we can substitute it back into equation (4) to find the value of a25:

a25 = 68 - 2(-68/9)

Simplifying the equation, we have:

a25 = 68 + 136/9

Finding a common denominator, we have:

a25 = (612 + 136)/9

Simplifying, we have:

a25 = 748/9

Therefore, the 25th term of the arithmetic series is 748/9.