Question

Calculati urmatoarea suma:
$S_{1} =1+2*5+3*5^{2} +4*5^{3} +...+30*5^{29}$
Ajutati-ma va rog frumos!!!

Answer 1

Răspuns

1*5^0+2*5^1+3*5²+4*5³+...+30*5^29=(115*5^30+5)/16

Explicație pas cu pas:

1+5+5²+5³+..+5^29=(5^30-1)/4

  5+5²+5³+    +5^29= 5(1+5+..+5^28)=5(5^29-1)/4

       5²+5³+.....5^29=...= 5²(5^28-1)/4

             5³+..+5^29=.....5³(5^27-1)/4

..............................................

                   5^28+5^29 =5^28(5²-1)/4

                               5^29=5^29(5-1)/4

S1=(1/4) (5^30-5+5^30-5²+5^30-5³+...+5^30-5^28+5^30-5^29)=

=(1/4)( 29*5^30-(5+5²+...+5^29))

dar  5+5²+5³+    +5^29= 5(1+5+..+5^28)=5(5^29-1)/4=(5^30-5)/4

deci

S1=(1/4)( 29*5^30-(5^30-5)/4))

S1=(1/4)(4*29*5^30-5^30+5)/4

S1=(1/16) *(116*5^30-5^30+5)=(115*5^30+5)/16