Personally, I would stay away from induction.
I was induced the day after my due date because "I was measuring big". I trusted my doctor, and didn't have easy access to the internet at that time to do any research on my own, so I just went along with whatever she wanted. Typically, induction is done with pitocin. I wound up going 10 hours with very little progress before they sent me home. During my induced labor, my blood pressure shot up. They waited for it to stabalize before going home. 3 days later I went into labor on my own, and went in. After several hours with slow progress, they decided to put me on pitocin again to speed things along. Once again, my blood pressure shot up only this time I developed pre-eclampsia. (And my pain level shot up too... Seriously, I was fine for one contraction, then they started the pitocin, and within a minute it was beyond my ability to handle. I snapped my arm in half (literally) and it doesn't compare to what pitocin labor was like. If you do get induced, get an epidural as well.) My labor still lasted another 12 hours after starting the pitocin, and after delivery I started hemmorhaging badly, and lost a lot of blood. A lot. My doctor told me afterward that the reason I bled so badly was likely due to the pre-eclampsia.
Well, after I went home I happened to read an article about inductions... It mentioned something about pitocin which piqued my curiosity and I looked up more information. Apparently, pitocin can cause high blood pressure, and women who are on it have higher rates of (drumroll) pre-eclampsia, eclampsia, and birth interventions.
Now, I'm not saying that pitocin is the devil, or that it should never be used. I'm not trying to "scare" you into NOT being induced... I know it has done a lot of good when used appropriately. I just don't think it should be used so casually. I think that inductions should be used when medically necessary, not because it's a convenient time for the doctor to deliver or because it "seems" like the time you should be giving birth.
Another interesting tidbit... (I read this somewhere, not sure about the source so don't take it as absolute fact...) Baby's lungs put off a hormone when they are fully developed and ready to go, which tells the body to go into labor.
7 days past due really isn't all that uncommon, btw. It's nearly impossible to pinpoint the exact time of conception, so the "due date" is just an educated guess, not a deadline. Unless there is an issue- low amniotic fluid, baby is stressed, your blood pressure is wonky, etc. I would really try to go into labor naturally.