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Answer:

Sulphur dioxide turns acidified potassium permanganate from purple to colourless.

Potassium permanganate, KMnO4, when used in titration, the initial oxidation of Mn is +7 and the color is deep purple. When the reaction is going on, it acts as a oxidizing agent and Mn gets reduced to the oxidation state of +2 . Thus turning the solution to colourless

Step-by-step explanation:

What will be the sequence generated by the generating function 4x/(1-x)2? d) 0, 1, 1, 3, 5, 8, 13,… Explanation: The sequence should be 0, 4, 8, 12, 16, 20,…for the generating function 4x/(1-x)2, when basic generating function: 1/(1-x). 8.

Answer:

If you take f(x)=1/(1 − x − x2) and expand it as a power series by long division, say,

then you get f(x)=1/(1−x−x2)=1+x+2x2 +3x3 +5x4 +8x5 +13x6 +···. It certainly

seems as though the coefficient of xn in the power series is the n-th Fibonacci number.

This is not difficult to prove directly, or using techniques discussed below. You can think

of the process of long division as generating the coefficients, so you might want to call

1/(1 − x − x2) the generating function for the Fibonacci numbers. The actual definition

of generating function is a bit more general. Since the closed form and the power series

represent the same function (within the circle of convergence), we will regard either one

as being the generating function.

Definition of generating function. The generating function for the sequence a0, a1,...

is defined to be the function f(x) = -

∞

n=0 anxn.

That is, the generating function for the sequence a0, a1,... is the function whose power

series representation has an as the coefficient of xn. We’ll call a0, a1,... the sequence

generated by f(x). We will not be concerned with matters of convergence, and instead

treat these as formal power series. Perhaps “symbolic” would be a better word than

formal.

When determining the sequence generated by a generating function, you will want to get

a formula for the n-the term (that is, for the coefficient of xn), rather than just computing

numerical values for the first few coefficients.

Useful facts.

• If x = 1 then 1+x+ x2 + ···+ xr = xr+1−1

x−1 (although it is often best not to do anything

with sums of only a few terms).

• 1

1−x =1+ x + x2 + ···.

Both of these facts can be proved by letting S be the sum, calculating xS − S, and doing

a little bit of algebra.

The second fact above says that 1

1−x is the generating function for the sequence 1, 1, 1, 1,....

It also lets you determine the sequence generated by many other functions. For example:

• 1

1−ax = 1+ a + a2x2 + a3x3 + ··· = -

∞

n=0 anxn. To see this, substitute y = ax

into the series expansion of 1

1−y

. Thus 1

1−ax is the generating function for the sequence

a0, a1, a2,.... The coefficient of xn is an.

1

1+ax = 1 − a + a2x2 − a3x3 + ··· = -

∞

n=0(−1)nanxn. To see this, substitute y = (−a)x

into the series expansion of 1

1−y

. Thus 1

1+ax is the generating function for the sequence

a0, −a1, a2,.... The coefficient of xn is (−a)n = (−1)nxn.

More useful facts.

• 1

1−x

k = -

∞

n=0 n+k−1

n

xn.

• 1

1−ax k = -

∞

n=0 n+k−1

n

anxn.

• 1

1+ax k = -

∞

n=0(−1)nn+k−1

n

anxn.

To see the first of these, consider the LHS, (1 + x + x2 + ···)k, multiplied out, but not

simplified, Each term is a product of k (possibly different) powers of x (remembering

that x0 = 1). By the rules of exponents, xn1 xn2 ··· xnk = xn1+n2+···+nk . Thus, after

simplifying, there is a term xn for every way of expressing n as a sum of k numbers each

of which is one of 0, 1, 2,.... That is, the number of terms xn equals the number of integer

solutions to n1 + n2 + ··· + nk = n, 0 ≤ ni, ∀i. We know that this number is n+k−1

k−1

(count using “bars and stars”).

To see the second fact, let y = ax in 1

1−y

k

. (This is the same idea as before.) To see the

third fact, let let y = −ax in 1

1−y

k

.

Multiplying a generating function by a constant multiplies every coefficient by that con-

stant. For example, 3

1+5x = 3 1

1+5x = 3(1 − 5x + 52x2 − 53x3 + ···)=3 − 3 · 5x + 3 · 52x2 −

3 · 53x3 + ··· = -

∞

n=0 3 · (−1)n5nxn. The coefficient of xn is 3 · (−1)n5n.

Multiplying a generating function by xk “shifts” the coefficients by k. This has the effect

of introducing k zeros at the start of the sequence generated. For example,

x

1+7x

4 = x4 1

1+7x

4

= x4 -

∞

n=0(−1)nn+4−1

n

7nxn

= -

∞

n=0(−1)nn+4−1

3

7nxn+4

= -

∞

t=4(−1)t−4t−1

3

7t−4xt

.

(Let t = n + 4 in the second last sum. If n = 0 then t = 4, so that gives the lower limit

for the sum. If n → ∞, so does t, which gives the upper limit. And n = t − 4.) Since

(−1)t−4 = (−1)t

, the last sum equals: -

∞

t=4(−1)t

t−1

3

7t−4xt

. The coefficient of xt is zero

if t < 4, and (−1)t

t−1

3

7t−4 if t ≥ 4.

Answer:

Madu went America under his mother .complete the blank using preposition phrases

Answer

The relationship could be, the more heated an object is there higher the temperature the object will have.

Explanation:

(Hope it helps)

Answer:

The heat of an object is the total energy of all the molecular motion inside that object. Temperature is the measure of the thermal energy or average heat of the molecules in a substance.

Explanation:

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