How about writing a letter to him 同易句下面的请问可不可以用Shall I write a letter to him

why not write a letter to him
Who don't you write a letter to him

除了你说的Shall I write a letter to him.
还有What about writing a letter to him.
Why not write a letter to him.
Why don't you write a letter to him.
以上是表示对对方做的事情征询看法
如果是自己做的事情,还有一种同义句型
let's /let me .okay ?
应该是总结全了