CPhill

 x+3y=7   subtract x from both sides

3y  = - x  + 7         divide both sides by 3

y = -x/ 3  + 7/3     and we can write this as

y = (-1/3)x  + 7/3

log₃ (27 · 81) and log₃ 27 + log₃ 81

The second is just the log expansion of the first......so we have

log₃ 27 + log₃ 81  =

3  +   4    =

7

3x  = 250 - x^2      add x^2 to both sides...subtract 250 from both sides

x^2 + 3x - 250   = 0      using the quadratic formula  and rejecting the negative value for x,  x  ≈ 14.382

I think you're confused about the "formula"

We have

a_n  = a_{(n - 1)}  + 2

So.....a₁   =  a_(1-1) + 2   =   a₀  + 2    =    -1  + 2   = 1

As written  we have  a + b / c = 2  ....   which   actually means    a   +   [ b / c]   =  2

Of course there could be infinite solutions, but one of them would be  if  a = 0 , b= -2  and c = -1  because

 0  + [ -2/ -1]    =   

0  +  2   = 

2

Another possibility is if a = -1, b = -6  and c = - 2

Notice that   -1 - 1 / -1  would actually give us ....   -1 -  (1/-1)   =  -1 - (-1)   =  -1 + 1   = 0

I'll try.....

a.     pH  = -log (2.3 * 10^-12)  ≈  11.6

b.    x  =  - log H+    where H+ is the concentration....so......writing this in exponential terms, we have......

-x = log H+   →     10^-x  =  H+

We are adding 2 to every successive term....so a₁ =   a₀  + 2 =     -1 + 2  = 1

There are 31 perfect squares from 1 to 1000

The sum of these = [ 31 (31 + 1)(2*31 + 1) ]  / 6   =  10,416      (1)

The sum of the first 1000 positive integers = (1000)(1001)/ 2 = 500,500    (2)

So....subtracting   (1) from (2)  should give us our answer :    500,500 - 10,416   = 490,084

Ah, yes....I mis-read the question.....my bad!!!!

You must leave by 9:30 AM   ...and that would be cutting it close!!!!