Find the sum of 35 terms of an arithmetic series of which the first term is "a" and the fifteenth term is "9a."

1st term --- a

15th term = a + 14d = 9a
14d = 8a
d = 4a/7

sum(35) = (35/2)(2a + 34d)
= (35/2)(2a + 34(4a/7)
= (35/2)(150a/7)
= 375a

To find the sum of the 35 terms of an arithmetic series, we need to know the value of the common difference (d) between the terms. The formula to calculate the sum of an arithmetic series is as follows:

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

Where:
- Sn represents the sum of the n terms
- a is the first term
- n is the number of terms
- d is the common difference between terms

In this case, we have the first term (a) and the fifteenth term (9a). Therefore, we can find the common difference (d) using the formula for the nth term of an arithmetic sequence:

an = a + (n-1) * d

Since we know that the fifteenth term (an) is 9a, we can substitute these values into the formula and solve for d:

9a = a + (15-1) * d
9a = a + 14d

Now, we can solve for d:

8a = 14d
d = 8a/14
d = 4a/7

Now, we have the values of a and d, so we can calculate the sum of the 35 terms (Sn):

Sn = (35/2) * [2a + (35-1) * d]
Sn = (35/2) * [2a + 34 * (4a/7)]
Sn = (35/2) * [2a + 136a/7]
Sn = (35/2) * [(14a + 136a)/7]
Sn = (35/2) * (150a/7)
Sn = (35 * 150 * a) / (2 * 7)
Sn = 5 * 150 * a
Sn = 750a

Therefore, the sum of the 35 terms is 750a.

To find the sum of the first 35 terms of an arithmetic series, we need to have information about either the common difference (d) or the first and last terms.

In this case, we know the first term (a) and the fifteenth term (9a). We can use this information to find the common difference (d) and then use the formula for the sum of an arithmetic series.

First, we need to find the common difference (d) by subtracting the first term (a) from the fifteenth term (9a):

9a - a = 8a

So the common difference (d) in this arithmetic series is 8a.

Next, we can use the formula for the sum of an arithmetic series:

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

where S is the sum of the series, n is the number of terms, a is the first term, and d is the common difference.

Plugging in the values we have:

S = (35/2)(2a + (35-1)(8a))

Simplifying:

S = (35/2)(2a + 34(8a))

S = 17(2a + 272a)

S = 17(274a)

S = 4658a

Therefore, the sum of the first 35 terms of this arithmetic series is 4658a.