Answer and explanation:
Please find attached answer and explanation
Answer:
Step-by-step explanation:
Hello, we want to prove that a proposition depending on n, that we can note P(n), is true for any n positive integer greater than 1. We need to follow several steps.
Step 1 - prove P(1)
For n = 1, n(2n+1)=1*3 =3 so we have
3 = 3, which is obviously true.
First step done!
Step 2 - for we assume P(k) and we need to prove P(k+1)
We assume that 3+7+11+...+(4k-1)=k(2k+1)
so we can write that
3+7+11+...+(4k-1)+(4(k+1)-1)=k(2k+1)+(4k+4-1)=k(2k+1)+4k+3
and
(k+1)(2(k+1)+1)=(k+1)(2k+3)
These two expressions are the same so it means that P(k+1) is true, meaning that
3+7+11+...+(4k-1)+(4(k+1)-1)=(k+1)(2(k+1)+1)
Step 3 - The conclusion
Finally, we have just proved that
3+7+11+...+(4n-1)=n(2n+1) for any n positive integer > 0
Thank you
Find the rate of change for the direct variation using the formula y = kx
Solve for k: 10 = k2
K = 10/2
K = 5
Now solve for y when x is 8:
Y = 5(8)
Y = 40
Answer:
y=x+15
Friend swims 40 minutes
Step-by-step explanation:
We re told that you want to swim 15 minutes <em>longer</em> than your friend. In word problems, longer or more always means addition. This means we will add 15 minutes to your friend's swim time (x) so we can figure out how long you swam (y).
This gives us the equation: y=x+15
Part two:
We know that you swam 55 minutes. This means y=55.
55=y=x+15
55=x+15
Now we want to get x alone, or isolate it. We can do this by subtracting 15 from both sides.
55-15=x+15-15
15-15= 0 and 55-15= 40
40=x
Thus your friend swam 40 minutes
(Remember that whatever you do to one side of the equation, you have to do to the other. If you add 2 to the left hand side, you have to add 2 to the right hand side. If you, like we did above, subtract 15 from the right, you have to subtract 15 from the left too)