Question

已知等比数列an,其前n项和为sn,已知s2=3a ,求s3/a3

Answer 1

问：已知等比数列an,其前n项和为sn,已知s2=3a ,求s3/a3

答：亲您好，根据等比数列的性质，可以得到：a1 * r = a2a2 * r = a3将上述两式相乘，得到：a1 * r * a2 * r = a2 * r * a3即 a1 * r^2 = a3又因为前n项和公式为：sn = a1 * (1 - r^n) / (1 - r)所以：s2 = a1 * (1 - r^2) / (1 - r)将s2 = 3a代入上式，得到：a1 * (1 - r^2) / (1 - r) = 3a化简得：a1 = 3r / (r + 1)将a1和a3代入s3的公式，得到：s3 = a1 * (1 - r^3) / (1 - r)a3 = a1 * r^2 = 9r^3 / (r + 1)^2所以：s3 / a3 = (a1 * (1 - r^3) / (1 - r)) / (9r^3 / (r + 1)^2)化简得：s3 / a3 = (r + 1)^2 * (1 - r^3) / (9r^2 * (1 - r))

答：由题目已知s2 = 3a，即：a1 * (1 - r^2) / (1 - r) = 3a化简得：a1 = 3r / (r + 1)^2代入s3 / a3的公式，得到：s3 / a3 = (r + 1)^2 * (1 - r^3) / (27r^2 * (1 - r))综上所述，s3 / a3 = (r + 1)^2 * (1 - r^3) / (27r^2 * (1 - r))，其中r为等比数列的公比。