Answer:
In Exercises 1-15 use mathematical induction to establish the formula for n ≥ 1.
1. 1
2 + 22 + 32 + · · · + n
2 =
n(n + 1)(2n + 1)
6
Proof:
For n = 1, the statement reduces to 12 =
1 · 2 · 3
6
and is obviously true.
Assuming the statement is true for n = k:
1
2 + 22 + 32 + · · · + k
2 =
k(k + 1)(2k + 1)
6
, (1)
we will prove that the statement must be true for n = k + 1:
1
2 + 22 + 32 + · · · + (k + 1)2 =
(k + 1)(k + 2)(2k + 3)
6
. (2)
The left-hand side of (2) can be written as
1
2 + 22 + 32 + · · · + k
2 + (k + 1)2
.
In view of (1), this simplifies to:
Answer:
(3/12)=0.25
(2/5)=0.40
(0.25/0.40)=0.625
Step-by-step explanation:
Start by completing the square.
x^2-8x-3y^2+12y=-16
x^2-8x+16-3(y^2-4y+4)=-16+16-12=-12
Notice the addition of those constants allows us to factor these expressions and because we did it to both sides it’s completely legit.
(x-4)^2-3(y-2)^2=-12
However, it seems we get a negative radius, which is not allowed. I think you entered the problem incorrectly.
This should be the correct answer,
4.5×10^9x + 0.00045x
Also, some good free calculators that can help with problems like this would be c y m a t h and m a t h w a y .
