Question

1st Empty box answer choices:

a) -4n^3+24n^2+12n
b) -4n^3+24n^2-12n
c) 4n^3+24n^2-12n

2nd Empty box answer choices:
a) 4n^4-24n^3+12n^2
b) -4n^4+24n^3+12n^2
c) 4n^4+24n^3+12n^2

Answer 1

The expression in the first empty box is -4n^3 + 24n^2 - 12n, and the expression in the second empty box is 4n^4 + 24n^3 + 12n^2.

Step-by-step explanation:

The expression in the first empty box is -4n^3+24n^2-12n. To determine this, imagine taking (n - 1) from the last term and adding it to the first term = 2[1 + (n - 1) + 3 + ... + (2n - 3) + (2n - 1) - (n - 1)] = 2[n + 3 + .... + (2n - 3) + n]. Now, let's look at the expression in the second empty box. The expression is 4n^4+24n^3+12n^2. To understand this, consider that the expression in the box is n². Imagine taking (n - 1) from the last term and adding it to the first term = 2[1 + (n-1) + 3 + ... + (2n - 3) + (2n - 1)-(n - 1)] = 2[n + 3 + .... + (2n - 3)+n]. Now take (n - 3) from the penultimate term and add it to the second term = 2[n + n + ... + n + n] = 2n².