10
find ∑ (k+2)*3*2^k
k=0
10∑k=0(k+2)⋅3⋅2k=3⋅10∑k=0(k+2)⋅2k=3⋅S10|We set S10=10∑k=0(k+2)⋅2k
S10=10∑k=0(k+2)⋅2k=2⋅20+3⋅21+4⋅22+⋯+11⋅29+12⋅2102⋅S10=2⋅21+3⋅22+⋯+10⋅29+11⋅210+12⋅211S10−2⋅S10=2⋅20+21+22+23+⋯+29+210−12⋅21120+21+22+23+⋯+29+210=211−121+22+23+⋯+29+210=211−2S10−2⋅S10=2⋅20+211−2−12⋅211(1−2)⋅S10=2⋅20+211−2−12⋅211−S10=211−12⋅211−S10=−211⋅(12−1)S10=211⋅11S10=11⋅211
10∑k=0(k+2)⋅3⋅2k=3⋅S10=3⋅11⋅211=33⋅211=67584
10
find ∑ (k+2)*3*2^k
k=0
10∑k=03(k+2)∗2k=3∗10∑k=0(k+2)∗2k=3[(2)+(3∗2)+(4∗4)+(5∗8)+(6∗16)+(7∗32)+(8∗64)+(9∗128)+(10∗256)+(11∗512)+(12∗1024)]
=67584
10
find ∑ (k+2)*3*2^k
k=0
10∑k=0(k+2)⋅3⋅2k=3⋅10∑k=0(k+2)⋅2k=3⋅S10|We set S10=10∑k=0(k+2)⋅2k
S10=10∑k=0(k+2)⋅2k=2⋅20+3⋅21+4⋅22+⋯+11⋅29+12⋅2102⋅S10=2⋅21+3⋅22+⋯+10⋅29+11⋅210+12⋅211S10−2⋅S10=2⋅20+21+22+23+⋯+29+210−12⋅21120+21+22+23+⋯+29+210=211−121+22+23+⋯+29+210=211−2S10−2⋅S10=2⋅20+211−2−12⋅211(1−2)⋅S10=2⋅20+211−2−12⋅211−S10=211−12⋅211−S10=−211⋅(12−1)S10=211⋅11S10=11⋅211
10∑k=0(k+2)⋅3⋅2k=3⋅S10=3⋅11⋅211=33⋅211=67584