tìm số tự nhiên n sao cho 2 × $2^{2}$ + 3 × $2^{3}$ + 4 × $2^{4}$ +…….+n × $2^{n}$ = $2^{n+34}$ 31/08/2021 Bởi Eliza tìm số tự nhiên n sao cho 2 × $2^{2}$ + 3 × $2^{3}$ + 4 × $2^{4}$ +…….+n × $2^{n}$ = $2^{n+34}$
Đáp án: 2A = 2.$2^{3}$ + 3. $2^{4}$ + 4. $2^{5}$ +… + n. $2^{n+1}$ -A=A-2A = 2.$2^{2}$ + (3.$2^{3}$ – 2.$2^{3}$) + …. +( n-n+1) $2^{n}$ – n. $2^{n+1}$ = 2.$2^{2}$ + $2^{3}$ + $2^{4}$+ …. + $2^{n}$ – n. $2^{n+1}$ => A = -2.$2^{2}$ – ( $2^{2}$ +$2^{3}$ + $2^{4}$+ …. + $2^{n+1}$) + (n+1). $2^{n+1}$ B = $2^{2}$+$2^{3}$ + $2^{4}$ + $2^{5}$ +… + $2^{n+1}$ 2B = $2^{3}$ + $2^{4}$ + $2^{5}$ +… + $2^{n+2}$ B= 2B- B= $2^{n+2}$ – $2^{2}$ => A = $2^{2}$ – $2^{n+2}$ -2.$2^{2}$ +(n+1). $2^{n+1}$ = (n+1). $2^{n+1}$ – $2^{n+2}$= (n+1-2). $2^{n+1}$= 2(n-1) $2^{n}$ => 2(n-1) = $2^{34}$ => n = $2^{33}$ + 1 Bình luận
Đáp án:
Giải thích các bước giải:
Đáp án:
2A = 2.$2^{3}$ + 3. $2^{4}$ + 4. $2^{5}$ +… + n. $2^{n+1}$
-A=A-2A = 2.$2^{2}$ + (3.$2^{3}$ – 2.$2^{3}$) + …. +( n-n+1) $2^{n}$ – n. $2^{n+1}$
= 2.$2^{2}$ + $2^{3}$ + $2^{4}$+ …. + $2^{n}$ – n. $2^{n+1}$
=> A = -2.$2^{2}$ – ( $2^{2}$ +$2^{3}$ + $2^{4}$+ …. + $2^{n+1}$) + (n+1). $2^{n+1}$
B = $2^{2}$+$2^{3}$ + $2^{4}$ + $2^{5}$ +… + $2^{n+1}$
2B = $2^{3}$ + $2^{4}$ + $2^{5}$ +… + $2^{n+2}$
B= 2B- B= $2^{n+2}$ – $2^{2}$
=> A = $2^{2}$ – $2^{n+2}$ -2.$2^{2}$ +(n+1). $2^{n+1}$
= (n+1). $2^{n+1}$ – $2^{n+2}$= (n+1-2). $2^{n+1}$= 2(n-1) $2^{n}$
=> 2(n-1) = $2^{34}$ => n = $2^{33}$ + 1