--> 测试数据：[tb]
IF OBJECT_ID('[tb]') IS NOT NULL DROP TABLE [tb]
GO 
CREATE TABLE [tb]([姓名] INT,[开始年] INT,[结束年] INT,[金额] INT)
INSERT [tb]
SELECT 1,2010,2011,20 UNION ALL
SELECT 1,2012,2014,10 UNION ALL
SELECT 2,2011,2011,10 UNION ALL
SELECT 2,2012,2012,10 UNION ALL
SELECT 3,2009,2010,10 UNION ALL
SELECT 3,2012,2013,10
--------------开始查询--------------------------
;WITH t AS(
SELECT *,isnull((select [开始年] FROM tb WHERE [姓名]=t.[姓名] AND ([开始年]=t.[结束年]+1  )),[开始年]) AS name  FROM [tb] AS t
)

SELECT [姓名],MIN([开始年]),MAX([结束年]),SUM([金额])
FROM t GROUP BY [姓名],name
ORDER BY [姓名]
----------------结果----------------------------
/* 
姓名    (无列名)    (无列名)    (无列名)
1    2010    2014    30
2    2011    2012    20
3    2009    2010    10
3    2012    2013    10
*/

------解决方案--------------------
--测试数据
IF OBJECT_ID('[tb]') IS NOT NULL DROP TABLE [tb]
GO  
CREATE TABLE [tb]([姓名] INT,[开始年] INT,[结束年] INT,[金额] INT)
INSERT [tb]
SELECT 1,2010,2011,20 UNION ALL
SELECT 1,2012,2014,10 UNION ALL
select 1,2015,2015,10 union all
select 1,2017,2017,10 union all
select 1,2018,2019,10 union all
select 1,2022,2022,10 union all
SELECT 2,2011,2011,10 UNION ALL
SELECT 2,2012,2012,10 UNION ALL
SELECT 3,2009,2010,10 UNION ALL
SELECT 3,2012,2013,10
GO
with a as(
select  
	*
	,case when not exists(select 1 from tb t2 where 姓名=t1.姓名 and t1.开始年=t2.结束年+1) then 1 else 0 end IsHeader  
	,case when not exists(select 1 from tb t2 where 姓名=t1.姓名 and t1.结束年=t2.开始年-1) then 1 else 0 end IsRooter 	
	,ROW_NUMBER()over(PARTITION by 姓名 order by 姓名,开始年) 姓名RN
from tb t1
),b as(
select  
	a1.*
	,(select (SUM(IsHeader+IsRooter)+1)/2  
	from a a2  
	where a2.姓名=a1.姓名 and a2.姓名RN<=a1.姓名RN) stage
from a a1
)
select  
	姓名,MIN(开始年) 开始年,MAX(结束年) 结束年,SUM(金额) 金额
from b
group by 姓名,stage
order by 姓名,stage
--写这段费死牛劲了...