Question

A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. Show your work.

Sn: 12 + 42 + 72 + . . . + (3n - 2)2 = (n(〖6n〗^2-3n-1))/2

Answer 1

my answer:
S1:1^2+4^2+7^2+…+(3*1-2)^2=1(6*1^2-3n-1)/2
S1: 1^2 = 1 (6*1^2-3*1-1))/2
S2: 1^2 + 4^2 = 2 (6*2^2-3*2-1))/2
S3: 1^2 + 4^2 + 7^2 = 3 ( 6*3^2 - 3*3 -1 ) /2 ...

and S4:1^2+4^2+7^2=(3*4-2)^2=4(6*4^2-3*4-1)/2

Sn : 1^2 + 4^2 + 7^2 + ... + (3n-2) ^2 = n ( 6*n^2 - 3n - 1) / 2 is true for all n, positive integers n=1,2,3,...

Answer 2

my teacher said:You need to show your work @hartnn @mathmath333 @master50777

Answer 3

@mathmath333 please check my Answer and what i did wrong here

Answer 4

\(\Large S_n: 1^2 + 4^2 + 7^2 + . . . + (3n - 2)^2 = \dfrac{(n(6n)^2-3n-1)}{2}\)

is this \(\Large S_n\)

Answer 5

yes

Answer 6

then ?@mathmath333

Answer 7

@mathmath333 then?

Answer 8

i think u have to use \( induction\) here

Answer 9

did ur techer asked u to write the proof by induction
or just
\(\Large s_1,s_2,s_3\)

Answer 10

she said :
8. You need to show your work. The letter S means this is a sum formula.
To do the proof, add all the terms together, show that it satisfies the formula for the values
of n given. So for n=1, S1=? and this should be equal to 1^2. For n=2, S2=? and this should be equal
to 1^2+4^2, etc. Please refer to pg. 952-959 for examples and sample problems.

Answer 11

i checked the above pages but culdnt get it

Answer 12

yes but in ur page 952-999 is their given proof by induction method

Answer 13

ok ill check it again

Answer 14

let move Q

Answer 15

Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit?

Answer 16

my answer : Grade A = 45 lbs
Grade B = 70 lbs
Total weight = A+B = 45+70 = 115
Blend Br = 1/3A + 2/3B
Blend Af = 1/2A + 1/2B

Profit = 1.50*lbs Blend Br + 2.00*lbs Blend Af

Profit = 1.50*(1/3A + 2/3B)+2,00(1/2A + 1/2B) =

45 pounds of A and 70 pounds of B yields max 24A + 72B = 98 Pounds Max Br
" " " " max 38A + 76B = 114 Pounds Max Af

The rate of profit for A is .50 per pound in BR and 1.00 per Pound in AF
For B, rate of profit for B is 1.00 for Br and 1.00 for Af.

Let X = # pounds in A
Let Y = # pounds in B

115 = A + B

A = 115-B

Let q = percent of Br
Let r = percent of Af

Let s = pounds of A in Br then 45-s = # pounds A in Af
Let t = pounds of B in Br and 70-t= # pounds in Af

.5s + 1.00*(45-s) + 1.00(t) + 1.00(70-t) = p

1.5s + 2.00t = p

If s = 45 pounds then 70 = 70/45 = 14/9s pounds

.5s + 1(45-s) + 1(14/9s) + 1(70-14/9s) = p

.5*45s + (45(1 - s)) + 14/9*45s + (45(1 - 14/9s) =p

22.5s + 45 - 45s + 70s + 35 - 70s = p

22.5s +70 - 115s = p

p = 137.5s + 70

p' = 137.5

22.5 A and 45 B for Br and 23.75 A and 23.75 B for Af.
Max profit is $137.5

its incomplete i guess or wrong? plz help
@mathmath333 @hartnn

Answer 17

i have no idea how u did that

here ia useful link

http://mathhelpforum.com/pre-calculus/202420-inequality-word-problem.html
here i found the equation while i cant get that how he formed
\(P=1.5x+2y\)

\(\large\tt \begin{align} \color{black}{
\dfrac{x}{3}+\dfrac{y}{2}\leq 45\\
\dfrac{2x}{3}+\dfrac{y}{2}\leq 70\\
x\geq 0\\y\geq 0
}\end{align}\)

u can graph it by linear programming
and find the
\(P_{max}\)

Answer 18

confusing :/

Answer 19

yes it confusing to me too

Answer 20

could you plz do it for me i tried already and got wrong i have to submit this before this year ends :(

Answer 21

m not also sureif the soln is right

see this graph

https://www.desmos.com/calculator/83t78nexye

u will find the region


with points
\((0,0),(0,90),(75,40),(105,0)\)

u have have to plug each of the value\in \(P(1.5x+2y)\)
and see and find the which is the maximum value of \(P\)

Answer 22

P (1.5(0) + 2(0)) P(1.5(0)+2(90)) P(1.5(75)+2(40)) P(1.5(105)+2(0))
P(0) P(180) P(4580) P(157.5)


P(4580), max value of P

Answer 23

\(\Huge \checkmark\)

Answer 24

\(p=4580\)