Math Is Fun Forum

You have it backwards.  You're solving x/10 = 25, when it's asking for 10/x = 25.

I made a spreadsheet to visualize the values and found this:

To determine if a specific n satisfies the conditions for a given x, find the unique prime factors of n+1 and call them p_1, p_2, ..., p_i.  If x+1 mod p_1, x+1 mod p_2, ..., x+1 mod p_i all do not equal 0 then n satisfies the conditions for x.

I'm not sure if this is helpful and I haven't worked a proof for it (although I can see how a proof could be done), but maybe it'll lead you in the right direction.

Edit:  A more useful way of stating the above would be this:

Given x, let the set A contain all prime numbers p_1, p_2, ..., p_i which are not factors of x+1.  Every n that satisfies the condition has the property that every prime factor of n+1 belongs to A.

You can get the correct answer by solving for the height, you just aren't being careful with what you're solving for.  When you solved for

you got 2/3 R, which is the radius.  However, if you solved

you probably got 1/3 R, right?  But remember that this answer is in terms of the height of the cylinder.  Remembering that b + h = R, you get b = R - 1/3 R = 2/3 R, which is what you got when you solved for the radius directly.

I believe #1 can be proven with the ratio test.  I'm probably going to butcher the terminology but this should be understandable.

The function A is equivalent to the series

Define a new function S

So A(x) = S(x).  Now the ratio test says that if

then S(x) will converge.  So what is

?  Well, it's

So we know that S(x) converges, which of course means A(x) does as well since they're equal.

I read #2 differently.  I think it's trying to say

That would make it far, far more difficult than #1 though, so I'm not sure.  In any case it's well beyond me so I'm afraid I can't help on that one.

Treat this formula as a function of n:

a, c, and d are constants that we need to solve for.  We are given 4 points on this function, though we only need 3 to solve it.  Using the given information we know that

You can see from here that a = 2, d = 2, and c = 1, so we're left with

To test this, try n = 4:

which is what we expected.  From here you can find any term you need.

You need to check 2 more cases.

CASE-IV: 9(11a+b) = 10x9
11a+b = 10 ⇒ a=1 and b=-1 Not Possible

CASE-V: 9(11a+b) = 19x18
11a+b = 38 ⇒ a=3 and b=5
a+b+c = 19
c = 19 - (a+b) = 19 - (3+5) = 11 Not Possible

If you're ok with a straight line it's pretty easy.  Let S be your starting value, T be your cumulative total, c be the number of additional values you're going to add, and x is the increment size.  This will give you the equation

For your example we have S = 1000, T = 7500, and c = 5, so

Well, you can narrow your search fairly easily.  There are two ways for your attribute to hold true: the two squares must have a difference of less than 100 or the unit's and ten's digits are equal.  For the first method there are only a finite number of possibilities since the distance between consecutive squares strictly increases.  Specifically, only the squares of 4 through 50 are eligible, and it's a pretty simple matter to brute force.  As it happens there is exactly one such pair of squares - 13 & 14.

For the second method you should note that in multiplication the final two digits of the product are decided solely by the final two digits of the multiplicands.  Again you can simply brute force numbers from 4 to 99 and check by hand.  For squares found using this method you can add any multiple of 100 and it will still work.  Aside from the obvious multiples of 10, the squares that fall under this category are 12, 38, 62, and 88.

In short, the numbers you are looking for are the squares of 10, 12, 13, 14, 20, 30, 38, 40, 50, 60, 62, 70, 80, 88, and 90, and any multiple of 100 plus any of those numbers except 13 and 14.

Your problem is missing something.  If you use x<-1 and an even value for n the inequality breaks down.

What you're really interested in is the length of a side of the pyramid at height z, since A(z) is just that number squared.  Let's call this length L(z), and note that it is simply a linear function.  You have L(0) = 0 and L(h) = w, so L(z) = zw/h, giving us A(z) = (zw/h)^2.  Integrate this over 0 to h and you get your answer.

It's been stipulated that M < 0, so