By the definition above,
and by substitution we get
If we keep following this procedure, we'll start to see a pattern:
and so on, down to
Now,
One of these series is geometric and has a well-known closed form:
The other (see note below) is a bit less obvious, but can be derived:
so we have
# # #
A brief note on how to compute the sum,
The first term of the sum is 0:
Add and substract 1 to the first factor in the summand and expand the sum into two sums:
The latter sum we're familiar with. For the other sum, we can shift the index to make it start at , and again the first term in the sum would be 0:
So we have
Continuing in this way, we would end up getting
or as a double sum,
which is easy to reduce:
A)
to solve a, the following rules are crucial:
(i)
so the difference of 2 logarithms with the same base, is the logarithm of their division, preserving the same base.
(ii) if
then b=c.
so if 2 logarithms with the same base are equal, then the arguments (b and c) are equal as well.
so
apply rule (i):
apply rule (ii):
now 'isolate' y:
b)
some more rules:
(iii)
(iv)
apply iii and iv:
then substitute
in the equation:
so now we have a quadratic equation of degree 2,
a=2, b=-5, c=2
the discriminant is
, the root of it is 3
so the roots are:
and
finally, we convert u's to x's:
means
so for u=2, x=4, and for u=1/2 we have x=1/4
Answers:
A) y=(6 x^{2} -1) \frac{(x-1)}{x}
B) solution set: {4, 1/4}
Answer:
22km
Step-by-step explanation:
As Sheila only walks 2.75 km one day out of the week, then she walks 2.75km/week. Because of this, the answer can be found by multiplying 2.75 by 8.
Answer:
what you can do with your point's
Step-by-step explanation:
What you can do with them is use them to make questions so say if it wanted to ask what 10+20x3-7=? then if someone answers it they will get the number of points you have added to the question.
The answer is D because the other answers are all correct.