若{an}和{bn}数列是等差数列,s,t为已知实数,求证{san+tbn}也是等差数列.
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若{an}和{bn}数列是等差数列,s,t为已知实数,求证{san+tbn}也是等差数列.
(san+tbn)-(san-1+tbn-1)
=(san-san-1)+(tbn-tbn-1)
=s(an-an-1)+t(bn-bn-1)
因为an-an-1为常数,bn-bn-1也为常数
所以s(an-an-1)+t(bn-bn-1)也为常数
所以{san+tbn}也是等差数列
=(san-san-1)+(tbn-tbn-1)
=s(an-an-1)+t(bn-bn-1)
因为an-an-1为常数,bn-bn-1也为常数
所以s(an-an-1)+t(bn-bn-1)也为常数
所以{san+tbn}也是等差数列
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