Hello, Godfather!

For a given arithmetic sequence the sum of the first fifty terms is 200

. . and the sum of the next fifty term is 2700.

What is the first term of the sequence?

The sum of the first n terms of an arithmetic sequence is given by:

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

The sum of the first 50 terms is 200.

. . S(50) .= .(50/2)[2a + 49d] .= .200 . → . 2a + 49d .= .8 .[1]

The sum of the next 50: (sum of the first 100) - (sum of the first 50)

. . (100/2)[2a + 99d] - 200

Hence, we have: .50[2a + 99d] - 200 .= .2700 . → . 2a + 99d .= .58 .[2]

Subtract [1] from [2]: .50d .= .50 . → . d = 1

Substitute into [1]: .2a + 49(1) .= .8 . → .a .= .-41/2