Find two positive nontrivial solutions of x^2-ny^2=1 when n is 15, 7, or 99

Hello,

if n=15: (4,1),(31,8),(244,63)

if n=7: (8,3),(127,48),(2024,765)

if n=99: (10,1),(199,20),(3970,399)

(Pell 's equation)

The solutions to the Pell equation

... x^2 -N·y^2 = ±1

are (x, y) = (p, q) where p/q are alternate convergents‡ of the continued fraction of √N.

The continued fractions for √15 and √99 can be written as

... √15 = 3 + 1/(1 + 1/(6 + 1/(1 + 1/(6 + 1/...

or, more compactly, as {3, {1, 6}};

and

... √99 = 9 + 1/(1 + 1/(18 + 1/(1 + 1/(18 + 1/...

or, more compactly, as {9, {1, 18}}.

The compact form of the continued fraction for √7 is {2, {1, 1, 1, 4}}.

In all cases, the continued fraction terms of a square root repeat after some finite-length sequence. In the compact form, the repeating sequence is shown in curly brackets. This permits the convergents to be defined recursively.

Successive convergents of the square root are p[k]/q[k], where

... p[k] = a[k]·p[k-1] +p[k-2] . . . . where p[-1]=0, p[0]=1

... q[k] = a[k]·q[k-1] +q[k-2] . . . . where q[-1]=1, q[0]=0

and the a[k] are written in the compact form as {a[1], {a[2], a[3], ...}}

For the given numbers N, the (x, y) values that are solutions to the given equation can be defined recursively quite nicely. The same recursive equation (written here as pq[ ]) is used for both p and q (as above), but with different initial conditions for p than for q.

For N=15, we can write

... pq[k] = 8·pq[k-1] - pq[k-2] . . . . . {p[-1], p[0]} = {4, 1} . . . {q[-1], q[0]} = {-1, 0}

so {x, y} = {4, 1}, {31, 8}, {244, 63}, {1921, 496}, ...

For N=99, we can write

... pq[k] = 20·pq[k-1] - pq[k-2] . . . . . {p[-1], p[0]} = {10, 1} . . . {q[-1], q[0]} = {-1, 0}

so {x, y} = {10, 1}, {199, 20}, {3970, 399}, {79201, 7960}, ...

For N=7, we can write

... pq[k] = 16·pq[k-1] - pq[k-2] . . . . . {p[-1], p[0]} = {8, 1} . . . {q[-1], q[0]} = {-3, 0}

so {x, y} = {8, 3}, {127, 48}, {2024, 765}, {32257, 12192}, ...

‡ Convergents alternate between being larger than the square root and smaller than the square root. However, when there are more than 2 values in the repeating sequence of the continued fraction, the convergents that result in a solution to the Pell equation may be farther apart than every other convergent. For N=7, every 4th convergent gives a solution.

i believe it is 0.1m because the measurement of 30.4m limits how precise you can be

you can only be as precise as the least precise measurement

(a) common ratio in the given sequence is, 3

(b) push ups will do kendall do on the 20th week

step-by-step explanation:

geometric sequence is a sequence of numbers that follows a pattern were the next term is found by multiplying by a constant called the common ratio(r)

(a)

given the sequence: 6 , 18 , 54 , 162 ,

this sequence is geometric sequence where each term is 3 times the previous term.

therefore,

common ratio(r) = 3.

(b)

a formula for nth term of geometric sequence is given by;

,   where r is the common ratio between the successive terms and is the first term,

to find how many push ups will kendall do on the 20th week.

here. n = 20th term.

in the given sequence first term

then;

simplify:

therefore, push ups will do kendall do on the 20th week