Question

What are your observations in the binomial expansion of (x+y)^n?

(a) The coefficients of the terms are symmetric.
(b) The sum of the coefficients of the terms is 1.
(c) The expansion contains n+1 terms.
(d) All of the above.

Answer 1

In the binomial expansion of (x+y)^n, the coefficients are symmetric, and the expansion contains n+1 terms, but the sum of the coefficients is 2^n, not 1.

Step-by-step explanation:

When we observe the binomial expansion of (x+y)^n, several characteristic patterns emerge. Firstly, the coefficients of the terms are indeed symmetric, which refers to the property that the ith term from the start has the same coefficient as the ith term from the end, for all i. This symmetry is due to the coefficients being derived from Pascal's Triangle, where each entry is the sum of the two entries above it. Secondly, the sum of the coefficients is not 1, but rather 2^n, which can be seen by substituting x and y each with 1 in the expansion (x+y)^n and understanding that it evaluates to 2^n. Lastly, there are indeed n+1 terms in the expansion. As an example, if n=3, the expansion (x+y)^3 will result in 4 terms: x^3 + 3x^2y + 3xy^2 + y^3.