Question

3¹+3²+.....+3⁶⁰ sec divide cu 52 (dau coroana)​

Answer 1

Am atasat o rezolvare.

Answer 2

Răspuns:

Explicație pas cu pas:

Progresie geometrica cu ratia q=3   b1=3    n=60

S=b1⋅(q^n-1)/(q-1)=3×(3^60-1)/(3-1)=3(3^30-1)(3^30+1)/2=

=3(3^15-1)(3^15+1)(3^30+1)/2=3(3^5-1)(3^10+3^5+1)(3^5+1)(3^10-3^5+1)(3^30-1)(3^30+1)/2=3×242×244×(3^10+3^5+1)(3^10-3^5+1)(3^30-1)(3^30+1)/2=

=3×242×244×59296×(3^10-3^5+1)(3^30-1)(3^30+1)/2=3×242×244×4561×13×

×(3^10-3^5+1)(3^30-1)(3^30+1)/2=3×121×244×4561×52×(3^10-3^5+1)(3^30-1)(3^30+1)/2

3^5=243  rezultatul se divide cu 2, cu 3,cu 4,  cu 13 ⇒ se divide si cu  52!

am folosit a²-b²=(a-b)(a+b) si a³+-b³=(a+-b)(a²+-ab+b³)