Let's start from what we know.

Note that:

(sign of last term will be + when n is even and - when n is odd).
Sum is finite so we can split it into two sums, first

with only positive trems (squares of even numbers) and second

with negative (squares of odd numbers). So:

And now the proof.
1) n is even.
In this case, both

and

have

terms. For example if n=8 then:

Generally, there will be:

Now, calculate our sum:



So in this case we prove, that:

2) n is odd.
Here,

has more terms than

. For example if n=7 then:

So there is

terms in

,

terms in

and:

Now, we can calculate our sum:




We consider all possible n so we prove that:
Answer:
48
Step-by-step explanation:
25% is 1/4 of the bag. 12x4 = 48.
There are 48 jelly beans in the bag.
Hope that helps
Answer:
Step-by-step explanation:
Begining of the month, value of stock=$78.35
At the end, the loss = $12.80
Value of stock at the end of the month= the difference between both values
$78.35 - $12.80 = $65.55
Answer:
y^3/(27 x^3)
Step-by-step explanation:
Simplify the following:
((3 x)/y)^(-3)
((3 x)/y)^(-3) = (y/(3 x))^3:
(y/(3 x))^3
Multiply each exponent in y/(3 x) by 3:
(y^3)/((3 x)^3)
Multiply each exponent in 3 x by 3:
y^3/(3^3 x^3)
3^3 = 3×3^2:
y^3/(3×3^2 x^3)
3^2 = 9:
y^3/(3×9 x^3)
3×9 = 27:
Answer: y^3/(27 x^3)
Answer:
A^2+B^2=C^2
Step-by-step explanation:
12=a
16=b
this is the pythagoreon theorem