1) Let S be the sum of a finite geometric series with negative common ratio whose first and last terms are 1 and 4, respectively. (For example, one such series is 1-2+4, whose sum is 3.)
There is a real number L such that S must be greater than L, but we can make S as close as we wish to L by choosing the number of terms in the series appropriately. Determine L.
2)If a/b rounded to the nearest trillionth is 0.008012018027, where a and b are positive integers, what is the smallest possible value of a+b?/
heres the problem
(a)Determine all nonnegative integers r such that it is possible for an infinite arithmetic sequence to contain exactly r terms that are integers. Prove your answer.
seems like an awefull lot of cases
(b)Determine all nonnegative integers r such that it is possible for an infinite geometric sequence to contain exactly r terms that are integers. Prove your answer.
Let x, y, and z be complex (i.e., real or nonreal) numbers such that x+y+z, xy+xz+yz, and xyz are all positive real numbers.
Is it necessarily true that x, y, and z are all real, and positive? If so, prove it. If not, give a counterexample.
Thanks. I can prove that they are positive if they are real numbers, but don't see the approach for if they are complex.
These are driving me mad:
(1)Suppose the polynomialhas integer coefficients, and its roots are distinct integers.
Given that a_n=2 and a_0=66, what is the least possible value of?
(2)Two of the roots of the polynomialhave product -32.
(3)Letbe the roots of f(x).
I just got another similar problem I thought was really cool:
Let r, s, and t be roots of the equation.
rs/t+st/r+tr/s = ((rs+st+tr)^2-2rst(r+s+t))/rst = (6^2-2*9*(-5))/9 = -6
but strangly, -6 = -(rs+st+rt)...
I dont see how that works
I just thought that was a really cool factorization.
so that can also be
(b^2+2ac)/c = a
maybe I discovered something new. Duno
For math class homework, I have several problems like this and I don't really know how to do these:
Let g(x) = x^4-5x^3+2x^2+7x-11, and let the roots of g(x) be p, q, r, and s.
(a) Compute pqr + pqs + prs + qrs.
Can you help me understand the method, not just do them for me?
(a) Give an example of two irrational numbers which, when added, produce a rational number. (sqrt(2) and -sqrt(2) )
Now let's consider just the addition of radicals.
(b) Suppose that a and b are positive integers such that bothare irrational. For what values of a and b is rational? Prove your answer.
(c) Again assuming a and b positive integers such that bothare irrational, for what values of a and b is rational? Prove your answer.
Thank you. I'm stumped!
on problem 3 I took
(p+q+r)^2 = 7^2
p^2+q^2+r^2 = 9
(p+q+r)^2-40 = p^2+q^2+r^2
(p^2+2 p q+2 p r+q^2+2 q r+r^2)-p^2-q^2-r^2 = 40
and 2 p q+2 p r+2 q r = 40
p q+p r+q r = 20,
but it askes for the average so the answer is 10, right?
well it says that is incorrect
1) Compute the sum
2) Find the ordered quintuplet (a,b,c,d,e) that satisfies the system of equations:
3) Suppose p+q+r = 7 and p^2+q^2+r^2 = 9. Then, what is the average (arithmetic mean) of the three products pq, qr, and rp?
4) Find the largest four-digit value of t such thatis an integer.