Explanations of serval different powers of (a+b)?

Question

eg: (a+b)^1, (a+b)^2

Answer 1

multiply it out. Of course, since this is the parapsychology

section, I will mentally project the rest of the answer to

telepathically.

Answer 2

(a + b)^1 = (a + b)

(a + b)^2 = a^2 + 2ab + b^2

(a + b)^3 = a^3 +3a^2b + 3ab^2 + b^3

(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4

Notice that the coefficients for the expansion in the first power are 1 and 1, that is, 1a + 1b. For the second, they are 1, 2 and 1 for 1a^2, 2ab and 1b^2. For the third they are 1, 3, 3, and 1, which I will write 1331. Then comes 14641, which are the coefficients for the fourth power of a+b, or (a+b)^4, as you can see above. The list can be continued as such:

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 21 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

If you write this as a pyramid, each number is the sum of the number above it and to the left with the number above it and to the right. Additional lines can be generated starting with 1 ans by adding consecutive terms. So, the next line would be

(1) (1+9) (9+36) (36+84) (84+ 126) etc. =

1 10 45 120 210 252 210 120 45 10 1

These would be the coefficients for the term (a+b)^10:

a^10 + 10 a^9b + 45a^8b^2 + 120 a^7b^3 ... +10ab^9 + b^10

Answer 3

What is your question?