If the sixth term of an arithmetic sequence is 5 and the sum of the third term and the eighth term is 5, what is the sum of the first nine terms of the arithmetic sequence?

If the sixth term of an arithmetic sequence is 5 and the sum of the third term and the eighth term is 5, what is the sum of the first nine terms of the arithmetic sequence?

Because this sequence is an arithmetic sequence, so S9 = 9A5
As long as we get A5, then the sum of the first nine terms of the sequence will be known
Because A3 + A8 = A5 + A6 = 5, and A6 = 5. So A5 = 0
So S9 = 9A5 = 0

It is known that the sixth term of the arithmetic sequence {an} is 5, and the sum of the third term and the eighth term is also 5

a6=a1+5d
a3+a8=a1+2d+a1+7d=2a1+9d=5
A1 = - 20
d=5
a9=-20+5*8=20
s9 = 1/2 * 9 * (-20+20) = 0

There are eight numbers, the average of which is 7.5. It is known that the average of the first four numbers is 9.5, and the average of the last five numbers is 8.8

Method 1: the average of the first four numbers is 9.5, the sum of the first four numbers is 38, and the average of the eight numbers is 7.5, so the average of the last four numbers is 5.5, the sum of the last four numbers is 22, the average of the last five numbers is 8.8, the sum of the last five numbers is 44, so the fourth number is 44-22 = 22