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The function is x²+1, and the product of the 97th term and the 98th term would be

This "n" would equal:

\(\sqrt{(97^2+1)(98^2+1)-1}\)

\(= \)\(\sqrt{97^298^2+98^2+97^2}\)

\(=\)\(\sqrt{90364036+9604+9409}\)

\(= \)\(\sqrt{90383049}\)

\(=\)\(9507\)

-Random Person :)

ahh ok i see, can you help me figure out my other question from my last post too that ones a bit difficult

THX CPhill, PS: happy belated B-Day

doesn't \(x = y = z = 2\)?

i didn't do it algebraically, but all of these equations are asking for "something" + "square of something" = 6, which for \(x = y = z = 2\) that results in

\(2 + 2^2 = 6 \).

the only other way i could think  to solve it logically is to think of the combinations that you can use to make 6, i.e. 1 + 1 x 6, 2 + 2 x 2. but the only values that satisfy all 3 equations are \(x = y = z = 2\) :) hope this helps

Thank you!

The largest 5-digit number is:

99972

Eh? What do you want help with?

Solve for a:
a^3 - a^2 = 4624

Subtract 4624 from both sides:
a^3 - a^2 - 4624 = 0

The left hand side factors into a product with two terms:
(a - 17) (a^2 + 16 a + 272) = 0

Split into two equations:
a - 17 = 0 or a^2 + 16 a + 272 = 0

Add 17 to both sides:
a = 17 or a^2 + 16 a + 272 = 0

Subtract 272 from both sides:
a = 17 or a^2 + 16 a = -272

Add 64 to both sides:
a = 17 or a^2 + 16 a + 64 = -208

Write the left hand side as a square:
a = 17 or (a + 8)^2 = -208

Take the square root of both sides:
a = 17 or a + 8 = 4 i sqrt(13) or a + 8 = -4 i sqrt(13)

Subtract 8 from both sides:
a = 17 or a = 4 i sqrt(13) - 8 or a + 8 = -4 i sqrt(13)

Subtract 8 from both sides:

a = 17 

I have not looked and there for do not know if these answers are correct. 

However this question has been answered before quite recently. If you search you can probably find the other answers.

What are you trying to do with the 3-year loan? Do you want to find the monthly payment? Just state your problem in plain English and will show you how to do it. OK ?

The centers  of the circles will form and equilateral triangle  each with a side of  2r

The  circumradius  of  this triangle  =     (2r) / √3

And the  circumcenter of this equilateral triangle  will be the  center of the larger circle

So.......the  radius  of the larger  circle =   r  + (2r)/√3  =    [ 3 + 2√3] r

                                                                                           _________             

                                                                                                  3

So we have that

3 + 2√3  =    [ 3 + 2√3] r

                   __________

                          3

1  =   r   / 3

r   =   3  =  radius of smaller circles

Here's a pic :  

DE  = radius of smaller circles

The number is:

17^3  -  17^2 =4,624

Yes n WAS the number of sucessful shots.

I probably should have used r because n is usually used for the total number of tries. - I have revised the original question now.

6 is your total number of tries.

-----------------------------

The formula for getting   r  sucesses in n independent trials its      (notice here i have used n to mean total number)

\(\color{red}\text{Prob (of r sucesses in n independent trials)}\\= \color{red}\;^nC_r\;*\;(\text{prob of a sucess)}^r\;*\text{(prob of a failure)}^{(n-r)}\)

In your questions

n=6

prob of sucess = \(\frac{211}{243}\)

Prob of failure = \(1-\frac{211}{243}=\frac{32}{243}\)

So, for instance in your example 

\(\text{Prob (2 sucesses in 6 independent trials)}\\= \;^6C_2\;*\;(\frac{211}{243})^2\;*\;(\frac{32}{243})^{(6-2)}\)

 log(log(log(log(x)))) = 0. 

Note that

log (1)  =  0

So.....this inplies that

 (log(log(log(x)))) = 1

And

log (10)  = 1

Which implies that

log (log (x) )   =   10

So   this implies that 

10^10  =  log x

x =  10^(10^10) = 10^(10000000000)

Let a  =  3   b  =  5  and  c  =  7

The circumradius is ginen by

             abc

_________________________________________  =

√[ (a + b + c) ( a + b - c) (a + c - b ) ( b + c - a)]

                3 * 5 * 7

_____________________________________  =

       √ [ ( 15) (1) ( 5) ( 9) ] 

 105

______  = 

15√3

  7

 __   =

√3

7√3

____

   3

sumfor(n, 1, 100, (n^(n+1) + n^(n-1))) mod 170 = 0

First one

A correlation coefficient  ot  -0.93   indictes a strong negative  correlation  between the variables

Second one

This data  is  widely scatterd and follows no linear pattern.....the correlation coefficient  is    0

Third one

This data plot indcates a strong negative  correlation between the variables......

Last one 

1 is correct  !!!!

Wait a minute !! I fouled the numbers up:

The smallest x, y, z are: 

x = 8,  y = 6,  z = 5

a=1; b=1;c=1;d=2^3*a + 2^4*b; if(d==2^5*c, goto5, goto6);printd,a,b,c; a++;if(a<100, goto3, 0);a=1;b++;if(b<100, goto3, discard=0; a=1;b=2;c++;if(c<100, goto3, 0)

The smallest values for x, y, z are:

x =2, y = 1 and z = 1

The maximum area will be that of an isosceles triangle in which you have the 3 sides =5, 5, 6.

Using Heron's formula for the area, you get:

Area = 12 sq. units.

Let A  = (0, 0)   B  =  (3,0)   

And we can let C  =  the intersection  of   circles with the equations  of

x^2 + y^2  =  25    and   (x -3)^2  + y^2  = 49

Subtract the first equation from the second and we have that

(x - 3)^2   - x^2   =   24   simplify

x^2 - 6x + 9  - x^2  =  24

-6x + 9  = 24

-6x  = 15

x = -15/6  =  -5/2  =  -2.5     this is the x coordinate of C

And the y coordinate  =     (-2.5)^2  + y^2  =  25

6.25 + y^2  =  25

y^2   =  25 - 6.25

y^2   = 18.75

y  = sqrt ( 75/ 4)

y = (5/2)√3  = 2.5√3

So...C  =    (-2.5, 2.5√3)

So....since M is the midpoint of BC....its coordinates are

( [ -2.5 + 3)/2 , [ 0 + 2.5√3] / 2 )   =   ( .25, 1.25√3)

So

AM  =   √[ .25^2  + (1.25√3)^2 ]  = √[ (1/4)^2 + (5√3/4)^2 ]  =  (1/4)√[ 1 + 75]  =  

√76  / 4    =   √[ 19 * 4 ] / 4    =   2√19/4    =  √19 / 2

Here's a pic :

Formula for circumradius is ("Multiply the three sides)"/4R = "The area of the triangle"

So using this formula we get 3* 5 * 7/ (4R) = 15sqrt3/4

So 105/(4R) = 15sqrt3/4

Dividing by 15 yields

7/(4R) = sqrt3/4

Multiplying by 4 yields

7/R = sqrt3

So R = 7sqrt3/3.

~~GF

HINT

I think you can solve this problem by using coordinates. Let BC be the x axis and let B have coordinates of (0,0). So C has coordinates of (7,0). You can find M by doing slopes. M has coordinates of (7/2, 0) because it is the midpoint. So AM can be found by using distance formula.

I will give the answer courtesy of a short computer code. Cphill will give an algebraic solution:

a=1; b=13;c= b-a; if(c==12, goto4, goto6);d=2#((b^2- a^2)/2 + a^2);printd,a,b; a++;b++;if(a<1000, goto2, 0)

The number are:
2^2 , 10^2, 14^2
4,      100,    196