Question

Rezolvand ecuatie 100x+356=S, unde S este suma termenilor sirului 6,13,20,27,...,706 obtinand x egal cu ?

Answer 1

$Termenii~sumei~au~forma~7k-1. \\ \\ S=(7 \cdot 1-1)+(7 \cdot 2-1)+ (7 \cdot3-1)+...+ (7 \cdot 101-1)= \\ \\ ~~~=7(1+2+3+...+101)-101= \\ \\ ~~~=7 \cdot \frac{101 \cdot 102}{2}-101= \\ \\ ~~~= 101( \frac{7 \cdot 102}{2}-1)= \\ \\ ~~~=101 \cdot356 \\ \\ 100x+356=S \Rightarrow x= \frac{S-356}{100}= \frac{101 \cdot 356-356}{100}= \frac{100 \cdot 356}{100}=356. \\ \\ \underline{Solutie}:~x=356.$