Cấp số nhân. - nâng cao

Home Lớp 11 Toán lớp 11 - Sách kết nối tri thức Bài 7. Cấp số nhân.Bài tập nâng cao

{"common":{"save":0,"post_id":"7053","level":3,"total":10,"point":10,"point_extra":0},"segment":[{"id":"4891","post_id":"7053","mon_id":"1158924","chapter_id":"1159147","question":"Hình vuông ABCD có c\u1ea1nh AB = $a$<\/span>, di\u1ec7n tích $S_1$<\/span> . N\u1ed1i 4 trung \u0111i\u1ec3m $A_1,B_1,C_1,D_1$<\/span> theo th\u1ee9 t\u1ef1 c\u1ee7a 4 c\u1ea1nh AB, BC, CD, DA ta \u0111\u01b0\u1ee3c hình vuông th\u1ee9 hai là $A_1B_1C_1D_1$<\/span> có di\u1ec7n tích $S_2$<\/span>. Ti\u1ebfp t\u1ee5c nh\u01b0 th\u1ebf ta \u0111\u01b0\u1ee3c hình vuông th\u1ee9 ba $A_2B_2C_2D_2$<\/span> có di\u1ec7n tích $S_3$<\/span> và c\u1ee9 ti\u1ebfp t\u1ee5c nh\u01b0 th\u1ebf ta \u0111\u01b0\u1ee3c di\u1ec7n tích $S_4,S_5,...$<\/span> . Tính $S=S_1+S_2+S_3+...+S_{100}$<\/span> .<\/p>","options":["A.<\/strong> $S=\\dfrac{2^{100}-1}{2^{99}.a^2}$<\/span>","B.<\/strong> $S=\\dfrac{a.(2^{100}-1)}{2^{99}}$<\/span>","C.<\/strong> $S=\\dfrac{a^2(2^{100}-1)}{2^{99}}$<\/span>","D.<\/strong> $S=\\dfrac{a^2(2^{99}-1)}{2^{99}}$<\/span>"],"correct":"3","level":"3","hint":"","answer":"Ta th\u1ea5y $S_1=a^2;S_2=\u200b\u200b\\dfrac{a^2}{2};S_3=\\dfrac{a^2}{4};...;S_{100}=\\dfrac{a^2}{2^{99}}$<\/span> .<\/p>Nh\u01b0 v\u1eady $S_1;S_2;S_3;...;S_{100}$<\/span> là c\u1ea5p s\u1ed1 nhân v\u1edbi s\u1ed1 h\u1ea1ng \u0111\u1ea7u $u_1=a^2$<\/span> và công b\u1ed9i $q=\\dfrac{1}{2}$<\/span> .<\/p>$S=S_1+S_2+...+S_100=a^2.\\bigg(1+\\dfrac{1}{2}+\\dfrac{1}{2^2}+...+\\dfrac{1}{2^{99}}\\bigg)$<\/span><\/p> $=a^2.\\dfrac{1-\\bigg(\\dfrac{1}{2}\\bigg)^{100}}{1-\\dfrac{1}{2}}$<\/span> $=\\dfrac{a^2(2^{100}-1)}{2^{99}}$<\/span><\/p>Ch\u1ecdn   C.<\/strong> $S=\\dfrac{a^2(2^{100}-1)}{2^{99}}$<\/span> .<\/span><\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2023-12-07 01:45:38","option_type":"math","len":0},{"id":"4893","post_id":"7053","mon_id":"1158924","chapter_id":"1159147","question":"Di\u1ec7n tích r\u1eebng t\u1ef1 nhiên c\u1ee7a n\u01b0\u1edbc ta n\u0103m 2021 là x\u1ea5p x\u1ec9  10 171 757 ha. Gi\u1ea3 s\u1eed t\u1ec9 l\u1ec7 t\u0103ng di\u1ec7n tích r\u1eebng t\u1ef1 nhiên hàng n\u0103m k\u1ec3 t\u1eeb n\u0103m 2021 là 0,7%\/n\u0103m. Di\u1ec7n tích r\u1eebng t\u1ef1 nhiên c\u1ee7a n\u01b0\u1edbc ta vào n\u0103m 2025 x\u1ea5p x\u1ec9 b\u1eb1ng<\/p>","options":["A.<\/strong> 10 386 863 ha","B.<\/strong> 10 532 788 ha","C.<\/strong> 10 468 251 ha","D.<\/strong> 10 459 571 ha"],"correct":"4","level":"3","hint":"","answer":"Di\u1ec7n tích r\u1eebng t\u1ef1 nhiên hàng n\u0103m k\u1ec3 t\u1eeb n\u0103m 2021 l\u1eadp thành m\u1ed9t c\u1ea5p s\u1ed1 nhân có công b\u1ed9i $q=1,007$<\/span> và s\u1ed1 h\u1ea1ng \u0111\u1ea7u là di\u1ec7n tích r\u1eebng t\u1ef1 nhiên n\u0103m 2021 là $u_1$<\/span> = 10 171 757 ha.<\/p>Do \u0111ó di\u1ec7n tích r\u1eebng t\u1ef1 nhiên c\u1ee7a n\u01b0\u1edbc ta vào n\u0103m 2025 là<\/p>$u_5=u_1.q^4$<\/span> = 10 171 757 . $(1,007)^4$<\/span> $\\approx$<\/span> 10 459 571 ha.<\/p>Ch\u1ecdn    D.<\/strong> 10 459 571 ha.<\/span><\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2023-12-07 01:54:45","option_type":"txt","len":2},{"id":"4895","post_id":"7053","mon_id":"1158924","chapter_id":"1159147","question":"Theo \u01b0\u1edbc tính, k\u1ec3 t\u1eeb lúc m\u1edbi mua, c\u1ee9 sau m\u1ed7i 200 l\u1ea7n s\u1ea1c thì pin c\u1ee7a \u0111i\u1ec7n tho\u1ea1i Iphone s\u1ebd gi\u1ea3m (chai pin) 5% so v\u1edbi chu kì 200 l\u1ea7n s\u1ea1c tr\u01b0\u1edbc \u0111ó. H\u1ecfi sau 1200 l\u1ea7n s\u1ea1c thì pin c\u1ee7a \u0111i\u1ec7n tho\u1ea1i Iphone còn l\u1ea1i bao nhiêu ph\u1ea7n tr\u0103m so v\u1edbi lúc mua?<\/p>","options":["A.<\/strong> 75%","B.<\/strong> 73,51%","C.<\/strong> 77,38%","D.<\/strong> 70%"],"correct":"2","level":"3","hint":"","answer":"Dung l\u01b0\u1ee3ng pin sau m\u1ed7i l\u1ea7n s\u1ea1c k\u1ec3 t\u1eeb lúc m\u1edbi mua l\u1eadp thành c\u1ea5p s\u1ed1 nhân có công b\u1ed9i $q=0,95$<\/span> và s\u1ed1 h\u1ea1ng \u0111\u1ea7u $u_1=100$<\/span>%.<\/p>Dung l\u01b0\u1ee3ng c\u1ee7a \u0111i\u1ec7n tho\u1ea1i Iphone sau 1200 l\u1ea7n s\u1ea1c (6 chu kì) còn l\u1ea1i so v\u1edbi lúc m\u1edbi mua là $u_7=u_1.q^6=100\\%.(0,95)^6\\approx 73,51\\%$<\/span> .<\/p>Ch\u1ecdn   B.<\/strong> 73,51% .<\/span><\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2023-12-07 02:01:09","option_type":"txt","len":0}]}