How do I solve these arithemic series math questions

What is the sum of all the positive multiples of 4 which are less than 100?

How many terms of the arithmetic series 95 + 90 + 85 +... are required to give a sum of 0?

If the first term of an arithmetic sequence is -9 and the common difference is 5, find the sum of the first 7 terms.

A useful property of arithmetic series is that the average of all the terms equals the average of the first and last terms.

1) The positive multiples of 4 which are less than 100 are 4, 8, 12, 16, ... , 96. This series has 96/4 = 24 terms and the average of the terms is (4 + 96)/2 = 50.
Therefore, the sum of the 24 terms is 24(50) = 1200.

2) Because we need the sum of the terms to be 0, we need the average to be 0. So the average of the first term (95) and the last term needs to be 0, so 95 plus the last term needs to be 0. Therefore, the last term must be -95.
Since the common difference is -5, there must be (-95 - (95))/(-5) + 1 = 39 terms.

3) The first term is -9 and the 7th (last) term is -9 + (7-1)(5) = 21. The average of all the terms is the average of -9 and 21, which is (-9 + 21)/2 = 6.
Therefore, the sum of the 7 terms is 7(6) = 42.

Lord bless you today!

1).
What is the sum of all the positive multiples of 4
which are less than 100?
4n (24 terms)
4 + 8 + 12 + 16 + .... + 96 = 1,200

2).
How many terms of the arithmetic series 95 + 90 + 85 +...
are required to give a sum of 0?

3).
If the first term of an arithmetic sequence is -9 and the common difference is 5,
find the sum of the first 7 terms.

2: 95/5 x 2 because the positive plus the negative will cancel each other out