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Just to clarify: the problem asks the sum of all the digits that he wrote down. 

sorry, but it says it's wrong... thanks for helping though :)

\(\boxed{72}\)

This is an arithemtic series with d = 1

S = (n/2) × (2a + (n−1)d)

S = sum = 999/2 (2 + 998(1))

               = 499500  

Using the formula \(\frac{1}{2}(n)(n+1)\), we substitute the \(n\) as \(999\) to get \(\boxed{499500}\).

Sorry, but I can't read your question clearly.

Ahhh thaks!

\(\boxed{3:1:5}\)

\(15 \cdot 14 \cdot 13 = \boxed{2730}\)

ohhh ok, that makes sense

"Non-integer"? Maybe you meant an "integer solution"?

y  = (x + 1)^x        take the ln of both sides

ln y  =  ln (x + 1)^x

ln y  =  x ln (x + 1)       differentiate both sides

y' / y  =  ln (x + 1)  + x (1/ (x + 1) )

y'  = y [ ln (x + 1)  + x/(x + 1) ]

y ' =  (x + 1)^x  [ ln (x + 1)  +  x /(x + 1) ]

We isolate x because we want the variables on the left hand side and the numbers on the right...

3x + 2 - 2 < 8 - 2

= 3x < 6

Note that this doesn't say greater than or equal to sign so 2 is not a solution...

1 is the only integer solution and the solution to the inequality is x < 2

Nice explanation CPhill!

Translating this into words it means a number multiplied by three is greater than 14 and less than 21. 5 and 6 are the only possible solutions to this inequality...

Well, you are told that you had this sequence: 1, 2, 2, 3, 3, 3. You can permute this sequence into:

6! / 2!.3! = 60 permutations, or sequences.

y =  log ₂ (8t ^ln2)

We can write

y = log ₂ 8  +  log ₂ t ^{ln 2}

y =   3  +  (ln 2) log ₂ t

y  = 3  + (ln2)  (log t / log 2)         

y =  3  + (ln 2)  ( ln t  / ln 2)

y  =  3  + ln t

y '  =  0  + 1/t  =   1/t

y =  3^(log ₂ t)

We can write

y =  3^ (  log t  / log 2 )   =  3^ ( ln t  / ln 2)

y '   =   3^( ln t / ln 2) * ln (3) * (1) / (t * ln 2)    = 3^ ( ln t / ln 2) * (ln 3)/(t ln 2)

Ducks, the total of the interior angles of any polygon is  (n – 2) • 180^o  where n is the number of sides. 

_.

Not sides.....the sum of the interior angles.....

To see this

A square is a quadrilateral

Each interior angle =  90

So....the sum of the interior angles =  4(90)  =  360    !!!!

So do all sides of the quadrilateral add up to 360?

Learning is fun!!!

This is a quadrilateral.....the  sum of its interior angles  = 360°

So

x + 2x  + 108 + 108  =  360

3x +216  =  360      subtract 216 from both sides

3x   =144     divide both sides by 3

x  = 48

Notice  that the diagonal of each of the yellow squares = sqrt (1^2 + 1^2)  = sqrt (2)

So....the width  of the yellow square at the top  = sqrt (2)

And if we draw a segment connecting  the bottom edges  of the two squares  directly adjacent to this top square we will form two (blue) right triangles  each with a base of   sqrt (2)  / 2    and a height of 1

So the area  of each of these triangles =  (1/2)( sqrt (2)/2) (1)  = sqrt (2)/4

And we have 8 such triangles ....so their total area  =  8 * sqrt (2) / 4   =  2sqrt (2)

And the remaining part  of the star  is just a square with a side length of sqrt (2)

So this area  =  (sqrt (2))^2  = 2

So....the total area of the blue star  =  2sqrt (2) + 2  =  2 (sqrt (2) + 1) ≈ 4.828 units^2

Here - "https://web2.0calc.com/questions/please-help_10631"

Of course this can't be totally accurate....but it appears to me that a  car that is 3 + 1/2 yrs. old  is worth about $3500

thankss! :)