Thread: Arithmetic Sequences

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?

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