1)Use Pascal's triangle to expand (w-x)^5

2)Use the binomial theorem to find the third term in the expression of (n-2p)^6

3)Which is NOT a counterexample to the formula 1^2+3^2+5^2+...+(2n-1)^2=n(2n+1)/3
A.n=3
B.n=2
C.n=1
D.n=4
is it C
thanks:)

For the binomial theorem, typing binomial theorem into Wikipedia should help from what I looked at on there.

1) To expand (w-x)^5 using Pascal's triangle, we can use the formula:

(w-x)^5 = C(5,0)w^5 + C(5,1)w^4(-x) + C(5,2)w^3(-x)^2 + C(5,3)w^2(-x)^3 + C(5,4)w(-x)^4 + C(5,5)(-x)^5

Pascal's triangle can be used to find the binomial coefficients for each term in the expansion. The binomial coefficients for the 5th row of Pascal's triangle are 1, 5, 10, 10, 5, 1, where the number 5 indicates the row number and the numbers within each row represent the coefficients.

Thus, the expanded form of (w-x)^5 is:

(w-x)^5 = 1w^5 + 5w^4(-x) + 10w^3(-x)^2 + 10w^2(-x)^3 + 5w(-x)^4 + 1(-x)^5

2) To find the third term in the expression (n-2p)^6 using the binomial theorem, we can use the formula:

(n-2p)^6 = C(6,0)n^6 + C(6,1)n^5(-2p) + C(6,2)n^4(-2p)^2 + C(6,3)n^3(-2p)^3 + C(6,4)n^2(-2p)^4 + C(6,5)n(-2p)^5 + C(6,6)(-2p)^6

Using Pascal's triangle, the binomial coefficients for the 6th row are 1, 6, 15, 20, 15, 6, 1.

Thus, the expanded form of (n-2p)^6 is:

(n-2p)^6 = 1n^6 + 6n^5(-2p) + 15n^4(-2p)^2 + 20n^3(-2p)^3 + 15n^2(-2p)^4 + 6n(-2p)^5 + 1(-2p)^6

The third term in this expansion is the term with the coefficient 15:

15n^4(-2p)^2 = 15n^4(4p^2) = 60n^4p^2

So, the third term in the expression (n-2p)^6 is 60n^4p^2.

3) To determine which value of n is NOT a counterexample to the formula 1^2+3^2+5^2+...+(2n-1)^2 = n(2n+1)/3, we can substitute each option into the formula and check if it holds true.

A) n = 3:
1^2 + 3^2 + 5^2 = 1 + 9 + 25 = 35
n(2n+1)/3 = 3(2(3)+1)/3 = 15

B) n = 2:
1^2 + 3^2 = 1 + 9 = 10
n(2n+1)/3 = 2(2(2)+1)/3 = 5

C) n = 1:
1^2 = 1
n(2n+1)/3 = 1(2(1)+1)/3 = 1

D) n = 4:
1^2 + 3^2 + 5^2 + 7^2 = 1 + 9 + 25 + 49 = 84
n(2n+1)/3 = 4(2(4)+1)/3 = 20

The formula holds true for n = 1, 3, and 4. Therefore, the NOT a counterexample to the formula is option B, n = 2.