Question

Sa se calculeze suma:
1+2*2+3*2^2+4*2^3+5*2^4+......+100*2^99.
Va rog mult sa ma ajutati!​

Answer 1

Răspuns:

Explicație pas cu pas:

99

∑     (k+1)* 2˄k  = ∑ k * 2˄k + ∑ 2˄k

k=0

∑ 2˄k = 1 + 2+2˄2+...+2˄99 = 2˄100 - 1

∑ k*2˄k = (n-1)*2˄(n+1)+2 se dem prin inductie

=98*2˄100+2

deci suma e 2 + 98*2˄100 + 2˄100 - 1 = 1+99*2˄100

astept verificare

Answer 2

Răspuns:

99*2^100+1

Explicație pas cu pas:

1+2+2²+2³+.....................................+2^99=2^100-1

  2+2²+2³+.........................+2^99=2(1+2+...+2^98)=2(2^99-1)=2^100-2

2²+..............................+2^99=2²(1+2+...+2^97)=2²(2^98-1)=2^100-2²

2³+..........................+2^99=2³(1+2+...+2^96) =2³(2^97-1)=2^100-2³

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

                             

2^98+2^99=2^98(2+1)=2^98(1+2)=2^98 *(2²-1) =2^100-2^98

                             2^99=2^99*1=2^99*(2-1)=  2^100-2^99  

   

toata suma =100*2^100-(1+2+2²+2³+...+2^98+2^99)=

=100*2^100-(2^100-1) =99*2^100+1