CPhill

Sorry for the slight error....of course the answer should be  100/101 = 1/1.01  = 0.9900990099009901

Note :  3/4+4/5+5/6+6/7+7/8+8/9  = 12611 / 2520

And : 4/5+5/6+6/7+7/8+8/9+9/10 = 12989 / 2520

So we have

log₃  [12611 / 2520 ] / log₁₀ [ 12989/2520]  which we can write as

[log [12611 / 2520] / log [3] ]  /  [ log [ 12989/2520] / log [10] ]       but log10  = 1....so we can ignore this....and we have

[log [12611 / 2520] / log [3] ]  /  [ log [ 12989/2520] ]

Unfortunately, this about all we can do  ......this evaluates to ≈ 2.058

1 + 3 + 5 + ..... + 199   can be expressed as the sum n^2  for n terms

The number of terms in this sum =  [ 1 + 199] / 2  = 200 / 2 = 100

So

1 + 3 + 5 + ....+199  = n^2 =  100^2  

And

 2 + 4 + 6 + ....+ 200    can be expressed  as the sum   n(n +1) for n terms

The number of terms in this sum  is the same as the first  = 100

So .... 2 + 4 + 6 + ....+ 200  =  n (n + 1)  = 100(101)  

So  

[1 + 3 + 5 + ....+199 ] / [ 2 + 4 + 6 + ....+ 200]  =  [ 100^2] / [ 100 * 101]  =

[100 * 100] / [ 100 * 101] =  1 / 101

sqrt(6-x)sqrt(x+7)=6x+8

sqrt(6 - x)  is only defined for x ≤ 6

sqrt (x + 7)   is only defined for x ≥ -7

But....since either of these would make the left side 0 - while the right side ≠ 0 -  we would have to confine ourselves to  x < 6 and x > -7......so.......

-7 < x < 6    and a, b =  -7, 6

Mmmmm....if I didn't know better, I might think that YOU could be our "Troll"  under a new alias.......

BTW - do you have a collection of monkeys that assist you ?????

Answered here :  http://web2.0calc.com/questions/are-these-answers-right

(a) Find the total cost function, and the total revenue function.

(b) How many dresses have to be sown and sold per month for her to break even?

(c) Find the profit function, and determine the profit when 90 dresses are sown and sold per month.

(d) How many dresses have to be sown and sold per month to yield a monthly profit of e4,500?

(a)  TC  = 1350 + 34U

       TR = 79U

(b) Breakeven

1350 + 34U  = 79U

1350 = 45U

U = 30

(c) Profit  = TR -TC  

79(90) - [1350  + 34(90)] = 2700

(d)  

4500  = 79U - [ 1350 + 34U]

4500  = 45U - 1350

5850 = 45U

U = 130

Here's a graph of the profit function with e break-even point  and the points in (c) and (d)  plotted  :

https://www.desmos.com/calculator/johprfhcwy

Just re-post them.......this sometimes happens....I don't know why......!!!!

1/7+2/7^2+1/7^3+2/7^4+...

Note that we can split this into two different sums

[1/7 + 1/7^3 +  1/7^5+......]  +  [ 2/7^2 + 2/7^4 + 2/7^6+....]

The sum of the first series =     [1/7] / [1 - 1/7^2 ]  = 7/48

And the sum of the second series  =  [ 2/7^2] / [ 1 - 1/7^2]  = 1/24

So.....the sum of the whole series =   7/48 + 1/24  =  7/48 + 2/48   =  9/48

A rectangle (square) with a length and width  of  40 meters

Proof  ......    2(w + L)   = 2 (40 + 40)  = 2 (80)  = 160 m