an arithmetic series is such that the ninth term is zero and the sum of first 25 terms is 50.the first term of the series?

just plug and chug:

a+8d = 0
25/2 (2a+24d) = 50

8d = -a, so

25/2 (2a-3a) = 50
a = -4

Thank you God bless you :)

To find the first term of the arithmetic series, we can use the formula for the nth term of an arithmetic series:

an = a1 + (n-1)d

Given that the ninth term (a9) is zero, we can substitute the values into the formula and solve for the common difference (d):

0 = a1 + (9-1)d
0 = a1 + 8d

Now, we can use the sum of the first 25 terms to find the common difference (d):

S25 = (n/2)(2a1 + (n-1)d)
50 = (25/2)(2a1 + (25-1)d)
50 = 12.5(2a1 + 24d)
4 = 2a1 + 24d

Now, we can solve the system of equations formed by the above equations:

0 = a1 + 8d
4 = 2a1 + 24d

We can multiply the first equation by 2 and subtract it from the second equation:

4 - 2(0) = 2a1 + 24d - 2(a1 + 8d)
4 = 2a1 + 24d - 2a1 - 16d
4 = 8d - 16d
4 = -8d
d = -0.5

Now, we can substitute the value of d back into the first equation to find a1:

0 = a1 + 8(-0.5)
a1 = 0 + (-4)
a1 = -4

Therefore, the first term of the arithmetic series is -4.

To find the first term of an arithmetic series, we can use the formula for the nth term of an arithmetic series:

an = a1 + (n - 1)d

where:
an is the nth term of the series
a1 is the first term of the series
n is the term number
d is the common difference between consecutive terms

Given that the ninth term is zero, we can substitute these values into the formula to find the common difference:

0 = a1 + (9 - 1)d
0 = a1 + 8d

Now, let's consider the sum of the first 25 terms of the series. The sum of an arithmetic series can be calculated using the formula:

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

where:
Sn is the sum of the first n terms
n is the number of terms

We are given that the sum of the first 25 terms is 50. Substituting the values into this formula, we have:

50 = (25/2)(a1 + 0)
50 = (25/2)a1
50 = 12.5a1

Now, we can solve for the first term (a1) by substituting the value of d we found earlier:

0 = a1 + 8d
0 = a1 + 8(50/12.5)
0 = a1 + 32
a1 = -32

Therefore, the first term of the arithmetic series is -32.