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\(\displaystyle \binom {10^9}{k} < \binom{10^9+1}{k-1}\\ \dfrac{(10^9)!}{k!(10^9-k)!}<\dfrac{(10^9+1)!}{(k-1)!((10^9+1)-(k-1))!}\\ \dfrac{1}{k(10^9-k)!} < \dfrac{10^9+1}{(10^9-k+2)!}\\ (10^9-k+1)(10^9-k+2) < (10^9+1)k\\ 1500000002-\sqrt{1250000003000000002} < k < 1500000002 + \sqrt{1250000003000000002}\)

Smallest positive integer k that satisfies the inequality is 381966012. (Used some computational intelligence :P)

10a + b = 38. :)

How would one calculate percent with ppm and promille, for example
What is 11% in parts per milion

I am not sure if there is something I do not understand in this question but I would say.

\(\frac{x}{1000000}=0.11\\ x=0.11*1000000\\ x=110000 \)

11% = 110000ppm

ok So Phil is going to do one exercise from each of 56 cards. 

Each card has just one exercise.

The only thing differnt between the workouts is the ORDER that the exercises are done in.

so there is   

56! ways that the cards can be ordered.   

56! =

=710998587804863451854045647463724949736497978881168458687447040000000000000 = 7.1099858780486345e74

so

56!   is about    7.1 x  10^74    ways to order the cards.     That is way more than 3 trillion orders.

1) - 180 =F * (1 - (7/4)^10) / (1 - 7/4), solve for F
F =47185920/93808891
b2+b4+b6+b8+b10
sum(82575360 / 93808891,  252887040 / 93808891,  774466560 / 93808891, 2371803840 / 93808891, 7263649260 / 93808891=1260 / 11

The sum of the first three terms of a geometric sequence of positive integers is equal to seven times the first term, and the sum of the first four terms is 45. What is the first term of the sequence?

We have that

7a₁ =  a₁ [ 1 - r^3] / [ 1 - r] 

7 [ 1 - r ] = 1 - r^3       factor the right as a differece of cubes

7 [ 1 - r] = [ 1 - r ] [ r^2 + r  +1]            divide out [ 1 - r ]

7 =   r^2 + r + 1    rearrange

r^2 + r - 6  = 0     factor

(r + 3) ( r - 2) = 0

Set each factor to 0 and solve for r  and we get that  r = - 3  or r = 2

Either

45 = a₁ [ 1 - (2)^4 ] / [ 1 - (2)]

45 = a₁ [ -15]/ (-1)

-45 = a₁ [-15]      divide by -15

3 = a₁           this must be the correct value of r since all terms are positive integers

8,000, -12,000, 18,000, -27,000, 40,500, -60,750, 91,125 =57,875.

In a geometric sequence of real numbers, the sum of the first six terms is 9 times the sum of the first three terms. If the first term is 5, what is the third term?

We have that

9* ( 5 [ 1 - r^3]   =  5 [ 1 - r^6 ]

9 [ 1 - r^3]  = 1 - r^6

9 - 9r^3 = 1 - r^6     rearrange as

r^6 - 9r^3 + 8 = 0           factor

(r^3 - 8) (r^3 - 1) = 0

r^3 - 1 = 0   has a positive solution of r = 1.....but r cannot take on this value

r^3 - 8 = 0    has a positive solution of  r = 2

So....the third term is  5(2)^2 = 20

Let S be the required score

The total of the two scores divided by two will give the average.

(S + 82) / 2 = 90     multiply both sides by 2

S + 82 = 180           subtract 82 from both sides

S = 180 - 82             complete the subtraction

S = 98                       Sarah must make a 98 on her next test to average 90

Thx for trying  ReallyWC. 

Thanks, ReallyWC....!!!!

[ 82 + x ] / 2 ≥ 90     multiply through by 2

82 + x ≥  180      subtract 82 from both sides

x ≥ 98

[She needs to make at least a "98" ]

Wasn't that your original answer, ReallyWC???

it was wrong.

Rearrange as

7 √ [ (2a)^2 + (1)^2 ]  - 4a^2 - 1   =  2 √[4a^2 + 1 ] + 6      rearrange again

7√ [ 4a^2 + 1 ] =  2 √[4a^2 + 1 ] + 6 + [4a^2 + 1]    rearrange again

[ 4a^2 + 1 ] - 5√ [4a^2 + 1 ] + 6 = 0

Now....Let   [4a^2 + 1 ]  =  m^2   ⇒  √ [4a^2 + 1 ]  = m     ......   and we have that

m^2 - 5m + 6 = 0         factor

(m -3) (m - 2) = 0

Setting each factor to 0  an solve for m and we get that

m = 2      or   m = 3

So

m^2 = 4  or m^2 = 9

If m^2 = 4  then we have that

4a^2 + 1 = 4

4a^2 = 3

a^2 = 3/4

a = ± √[3] /2

If m = 9,then we have that

4a^2 + 1 = 9

4a^2 = 8

a^2 =8/4

a^2 = 2

a =  ± √2

So.....the largest value for a is √2

xxxxxx.

You question needs to be fixed. You have used the wrong letter, the first pqrs should be abcd.

a   2 = (1.11)^x

      log2/log1.11 = x = 6.64 yrs

b  2 =( 1+.11/12)^12t

     log2/log1.009166) = 12t     t = 6.33 yrs

c  2= (1 +.11/365)^365t

       t = 6.3022 yrs

d   2 = e^(.11)t      t = 6.3013 yrs

   As the compounding period becomes smaller the amount of time becomes shorter (i.e.  the APY (annual Percentage Yield) becomes more)

1 1/4 =    125 %

125 %  of 'x' = 38000     where 'x' is the business degree starting salary

1.25 x  = 38000

x = 38000/1.25 = $ 30400

Only if you divide by a negative number.

a) 6.64188 years

b) 6.64188 months

c) 6.64188 days

d) you should be able to figure this out

see a trend yet?

HANG ON HERE! when i divided i could switch the sign from < to >. isnt that how you do it?

\(3x + 3 (x-1) < 7 - 2x + 6\\ 3x + 3x-3 < 13 - 2x\\ 6x-3 < 13 - 2x\\ 8x<16\\ x<2 \)

I would think x<2.

really? that is crazy. math is so cool.