523/HK17. problem / Math - Solved Math Problems

MR-523 / 8. problem

The fourth member of the arithmetic series is 7, and the fifth is -5.
Determine the first member of the series!

Source: Magyarország: Unified final exam for high schools - Mathematics - intermediate level [09.05.2017.] - 8..)

Algebra → Exponential equations and inequalities → Exponential equations

a_{4} = 7, a_{5} = - 5

(1) a_{4} = a_{1} + 3 d = 7

(2) a_{5} = a_{1} + 4 d = - 5

(1) a_{1} + 3 d = 7

(2) a_{1} + 4 d = - 5

(2) - (1) (a_{1} + 4 d) - (a_{1} + 3 d) = - 5 - 7

$(2) - (1) a_{1} + 4 d - a_{1} - 3 d = - 5 - 7$

$d = - 12$

(1) \to a_{1} = 7 - 3 d

a_{1} = 7 - 3 (- 12) = 7 + 36

$a_{1} = 43$