设数列b‹n›=(2n -1)3ⁿ,求{b‹n›}的前n项和为T‹n›
T‹n›=1×3+3×3²+5×3³+7×3⁴+.+(2n-3)×3ⁿ⁻¹+(2n-1)×3ⁿ.(1)
3T‹n›=1×3²+3×3³+5×3⁴+7×3⁵+.+(2n-3)×3ⁿ+(2n-1)×3ⁿ⁺¹.(2)
(1)-(2)【错项相减】得：
-2T‹n›=1×3+2(3²+3³+3⁴+3⁵+.+3ⁿ)-(2n-1)×3ⁿ⁺¹
=3+6(3¹+3²+3³+3⁴+3ⁿ⁻¹)-(2n-1)×3ⁿ⁺¹
=3+6[3(3ⁿ⁻¹-1)/2]-(2n-1)×3ⁿ⁺¹
=3+9(3ⁿ⁻¹-1)-(2n-1)×3ⁿ⁺¹
=3+3ⁿ⁺¹-9-(2n-1)×3ⁿ⁺¹
=-6-2(n-1)×3ⁿ⁺¹
∴T‹n›=3+(n-1)×3ⁿ⁺¹