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Yeah I think I did the same the first time I solved it

I think I got the same answer when I did it before....I believe I omitted one of the leap years....!!!

Thanks CPhill! I got 243 and did 249-6 instead of 249-7! Now I understood my really silly mistake :)

2999 - 2001  =  998 years

Number of possible leap years  =  floor (998 / 4)  = 249 years

But    2100    2200   2300  2500  2600  2700  2900 will not be evenly divisible by 400

So

249 - 7  =   242 leap years

Thanks CalculatorUser and CPhill!

WolframAlpha shows a solution of x = 105

Maybe a substitution could solve this??? [ But.....I'm not sure ]

Anyone know an algebraic solution  ????

Very nice, CU  !!!!!

(b)   Similarly

2(3) + t = 0

6 + t = 0

t = - 6

And

(r) / (2) = -4

r = 8

And if we have (1, 0)   then

r(1) + s = 0

8(1) + s = 0

s = -8

So    (r, s, t )  =  ( 8, -8, -6)

a)  If the vertical asymptote  = 3

Then   this value of x makes the denominator = 0

So......3 + c  = 0 ⇒  c = -3

If we have a horizontal asymptote of - 4...this implies that (ax/1x) = (a / 1) =  - 4  ⇒  a = -4

And if the x intercept =   then   a(1) + b = 0  ...so...

-4(1) + b = 0

-4 + b = 0

b = 4

So   (a, b , c ) =  ( -4, 4, -3)

Thanks CPhill!

A little confusing, but....

Term  number   1   2    3     4     5      6     7

                          3  7   15    27   43   63   87

Note that the 3rd term  = previous term + 4 times the previous term number  = 7 + 4(2) = 15

4th term  =  15 + 4(3)  = 27

5th term = 27 + 4(4)  = 43

6th term =  43 + 4(5) = 63

7th term = 63 + 4(6) = 87

Notice something else.....the difference between terms is   4, 8, 12, 16, 20, 24

Just for the heck of it, what would be the 8th term   ????

Since \(\sqrt{n}\) can be 5,6,7,8,9.

You sqaure the possible solutions to get 25,36,49,64,81.

These are the ONLY solutions because the problem said "positive integers"

So there are 5 possible solutions

So since 7 is the second term, we multiply 2 by 4 and add it to 7. We get 15

Our sequence so far: 3,7,15,

Then we multiply 4 and 4 to get 16 as 15 is the FOURTH term then add 16 to 15. We get 31.

Do you notice a pattern? 3,7,15,31.

Yes 2n+1 can represent each change for the terms.

So we keep calculating...

Each successive number is "two times three less"   means  "six less"    

So 

1.   0 , 6, -12 , -18 , -24

Can you do the other two, GM, knowing this  ????

[ Hint for number 3.....start with  an integer > 24 ]