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 .

your mom x 3

step-by-step explanation:

the cat is blue because the probability of extension towards the prefix of the diameter of the circle could only mathematically equal to the sum of the roots of the carrots from   your garden.

step-by-step explanation: