Hello.
How to prove that those operations are truth?
Sums

a. This is just shifting the index of summation:

and just swap j with k, since you can freely use any symbol for the index.

That is, the identity relies on simple integer arithmetic: 0 + 1 = 1 = 2 - 1, and 1 + 1 = 2 = 3 - 1, and so on.

b. This is something of a generalization of the identity in part a.

c. Distribute the summation to get

The second sum here is just adding together m copies of 1,

so

d. The sum as given is generally not true:

while (generalizing the identity from part c)

and these expressions are equal only if .

518,400

step-by-step explanation:

the area is multiplying all the given measurements together.

40 * 18 * 30 * 24 = 518,400

3: 2

step-by-step explanation:

shorter piece is 2: 3, so just reverse it, and you get 3: 2