在数学上,等差-等比数列(简称差比数列,英语:arithmetico-geometric sequence)是一个等差数列与一个等比数列相乘的积。
等差-等比数列有如下通项公式:[1]
![{\displaystyle [a+(n-1)d]r^{n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c98b4f9faa620ac0b0b6f32e09cae6ee75216b9)
其中
是公比,而
的系数:
![{\displaystyle [a+(n-1)d]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fdade0a828ef6c5fafd1ad31e1333e1f8fdf4a5)
则是等差数列的项,其首项为
,公差
。
等差-等比级数有如下形式;
![{\displaystyle \sum _{k=1}^{n}\left[a+(k-1)d\right]r^{k-1}=a+(a+d)r+(a+2d)r^{2}+\cdots +[a+(n-1)d]r^{n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cb778ef20d37ad57e0f9188ad365ad17de866d8)
其前n项之和为;
![{\displaystyle S_{n}=\sum _{k=1}^{n}\left[a+(k-1)d\right]r^{k-1}={\frac {a}{1-r}}-{\frac {[a+(n-1)d]r^{n}}{1-r}}+{\frac {dr(1-r^{n-1})}{(1-r)^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d467c0294256a6823d2885db162224ba0951072)
由此级数开始:[1][2]
![{\displaystyle S_{n}=a+(a+d)r+(a+2d)r^{2}+\cdots +[a+(n-1)d]r^{n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce10c2a23b8dbc6db1b31c312f470fc04194032)
将Sn乘以r,
![{\displaystyle rS_{n}=ar+(a+d)r^{2}+(a+2d)r^{3}+\cdots +[a+(n-1)d]r^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7bace018677fc5840cfae99964ec8b5358b693e)
Sn减去rSn,
![{\displaystyle {\begin{aligned}S_{n}(1-r)&=&\left\{a+(a+d)r+(a+2d)r^{2}+\cdots +[a+(n-1)d]r^{n-1}\right\}\\&&-\left\{ar+(a+d)r^{2}+(a+2d)r^{3}+\cdots +[a+(n-1)d]r^{n}\right\}\\&=&a+\left[dr+dr^{2}+\cdots +dr^{n-1}\right]-[a+(n-1)d]r^{n}\\&=&a+\left[{\frac {dr(1-r^{n-1})}{1-r}}\right]-[a+(n-1)d]r^{n}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d158643292018dc1ad6e84c188ef0ba85ab7e5b)
在中间的项中使用等比数列的求和公式。最后左右两边同除以(1 − r),得到最终结果。
![{\displaystyle \displaystyle \sum _{k=1}^{n}r^{k-1}={\frac {r^{n}-1}{r-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d48bc1180809a7649a8edea0070866a4c54e70f0)
对等比数列和两边求导:[3]
![{\displaystyle \displaystyle \sum _{k=1}^{n}(k-1)r^{k-2}={\frac {nr^{n-1}}{r-1}}-{\frac {r^{n}-1}{(r-1)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/650be71447860258bfb7eea38343163690a25c34)
![{\displaystyle \displaystyle \sum _{k=1}^{n}[a+(k-1)d]r^{k-1}=a{\frac {r^{n}-1}{r-1}}+dr[{\frac {nr^{n-1}}{r-1}}-{\frac {r^{n}-1}{(r-1)^{2}}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/249b87c7b48b45715959865d069306b1caefef40)
待定系数s,t使得等差-等比数列可以裂项:[4]
![{\displaystyle [a+(k-1)d]r^{k-1}=(sk+t)r^{k}-[s(k-1)+t]r^{k-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bab1a711d703256fc78e31218ea8e556dbc4ade2)
用裂项法可以求出数列和:
![{\displaystyle \displaystyle \sum _{k=1}^{n}[a+(k-1)d]r^{k-1}=(sn+t)r^{n}-t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40c9cba3e04c8d4bb3a21f2ff350c85657556626)
求出待定系数s,t关于a,d,r的表达式:
![{\displaystyle dk+a-d=s(r-1)k+(r-1)t+s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a938cbb8be02538159a792d2345fa03adb1d291)
![{\displaystyle \displaystyle s={\frac {d}{r-1}},t={\frac {a-d}{r-1}}-{\frac {d}{(r-1)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7a7b935b4337032a89241e4258bfd3cb4ebe0a8)
![{\displaystyle \displaystyle \sum _{k=1}^{n}[a+(k-1)d]r^{k-1}=[{\frac {d}{r-1}}n+{\frac {a-d}{r-1}}-{\frac {d}{(r-1)^{2}}}]r^{n}-[{\frac {a-d}{r-1}}-{\frac {d}{(r-1)^{2}}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c45bcfeb0b578847d2748f77336e760adf54602e)
[5]
- 其中
![{\displaystyle \Delta p(n)=p(n+1)-p(n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e85dbd274733b76cfa698eeefcb0cfa65128e434)
求出各阶差分:
![{\displaystyle \displaystyle f(n)={\frac {a+(n-1)d}{r-1}}-{\frac {d}{(r-1)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c2bcb63b5270fab10b945268e80b1cd45d131a)
![{\displaystyle \displaystyle \sum _{k=1}^{n}[a+(k-1)d]r^{k-1}=[{\frac {a+(n-1)d}{r-1}}-{\frac {d}{(r-1)^{2}}}]r^{n}-[{\frac {a-d}{r-1}}-{\frac {d}{(r-1)^{2}}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc72cab8df2afc68e6f8c06531dca45d8ad47d68)
如果
,那么其无穷级数为[1]
![{\displaystyle \lim _{n\to \infty }S_{n}={\frac {a}{1-r}}+{\frac {dr}{(1-r)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c33dc988f0bf4e7bfa5dc08649de97f98581f6d3)
如果
在上述范围之外,则该级数不是发散级数就是交错级数。