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Determine a formula involving binomial coefficients for the nth term of a sequence if its initial terms are those listed. [Hint: Looking at Pascal’s triangle will be helpful. Although infinitely many sequences start with a specified set of terms, each of the following lists is the start of a sequence of the type desired.]

1. $1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66,\dots$
2. $1, 4, 10, 20, 35, 56, 84, 120, 165, 220,\dots$
3. $1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620,\dots$
4. $1, 1, 2, 3, 6, 10, 20, 35, 70, 126,\dots$
5. $1, 1, 1, 3, 1, 5, 15, 35, 1, 9,\dots$
6. $1, 3, 15, 84, 495, 3003, 18564, 116280, 735471, 4686825,\dots$