Upvote:5
First rule of dealing with immigration officers: never lie. Never, ever, ever lie. When applying for the visa, include the invitation letter and add that you want spend one more week. Same at the border. It's perfectly normal and understandable that you want to spend a week on holiday after a business trip. I can't even count how many times I answered to the border official "both" when asked this question -- I am going to the conference and visiting friends. It's fine.