如果数列{an}满足a1,a2/a1,a3/a2,...an/an-1,...是首项为1,公比为2的等比数列,则a101
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如果数列{an}满足a1,a2/a1,a3/a2,...an/an-1,...是首项为1,公比为2的等比数列,则a101等于()
a1 = 1;
a2/a1 = 2; => a2 = a1*2; => a2 = 2;
a3/a2 = 4; => a3 = a2*4; => a3 = 8;
.
证明:
因为 ai/a(i-1) = 2^(i-1); (i>=2)
=> ai = 2^(i-1)* a(i-1)
=> ai = 2^(i-1)*2^(i-2)*a(i-2);
.这里可以顺次展开,一直展开到2^(1)*a1;得下式
=> ai = 2^(i-1)*2^(i-2)*.*2^(i-(i-1))*a1;
=> ai = 2^( i*(i-1)/2);
a2/a1 = 2; => a2 = a1*2; => a2 = 2;
a3/a2 = 4; => a3 = a2*4; => a3 = 8;
.
证明:
因为 ai/a(i-1) = 2^(i-1); (i>=2)
=> ai = 2^(i-1)* a(i-1)
=> ai = 2^(i-1)*2^(i-2)*a(i-2);
.这里可以顺次展开,一直展开到2^(1)*a1;得下式
=> ai = 2^(i-1)*2^(i-2)*.*2^(i-(i-1))*a1;
=> ai = 2^( i*(i-1)/2);
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