I need help with this math problem

GP ​

We have

(a + bx) / (a - bx) = (b + cx) / (b - cx)

==>   (a + bx) (b - cx) = (a - bx) (b + cx)

==>   ab + (b ² - ac) x - bcx ² = ab + (ac - b ²) x - bcx ²

==>   (b ² - ac) x = (ac - b ²) x

==>   b ² - ac = ac - b ²

==>   2b ² = 2ac

==>   b ² = ac … … … [1]

Similarly, you would find

(a + bx) / (a - bx) = (c + dx) / (c - dx)

==>   ad = bc … … … [2]

and

(b + cx) / (b - cx) = (c + dx) / (c - dx)

==>   c ² = bd … … … [3]

Now:

c ² = bd   ==>   b = c ² / d

b ² = ac   ==>   c = b ² / a

ad = bc   ==>   d = bc / a

and we find

d / c = (bc / a) / (b ² / a) = c / b

and

c / b = (b ² / a) / (c ² / d) = (b ² d) / (a c ²) = b / a

which is to say, the ratio between d and c is equal to the ratio between c and b, and also equal to the ratio between b and a. Therefore (a, b, c, d) are in a geometric progression.

step-by-step explanation: so we look at the formula un = a + (n-1)d

un is the amount of seats

n is the nth term aka row number

a is the original sequence/row number

d is the difference between each term

so in this a=22 d=6 and n can remain as n because we are simply formjng a rule

lets put these in the equation

un=22+(n-1)(6)

this is your answer to part a

however

lets double check that this is correct, we'll try putting it into the second term

un=a+(n-1)d

u2=22+(2-1)(6)

u2=22+6

u2=28

as the question said the second row has 28 seats so now we know the formula we made is correct.

as for part b

un=a+(n-1)

its asking us to find a row with 100 seats.

remember un was the amount of seata so all we do is substitute this in.

un=100 a=22 d=6

un=a+(n-1)d

100=22+(n-1)(6)

so now we simplify to find n which is the row number.

100-22=22+(n-1)(6)-22

78=(n-1)(6)

78/6=(n-1)(6)/6

13=n-1

13+1=n-1+1

14=n

so now we know that in the 14th row there is 100 seats.

hope this and that you understood! its my first time using the app or anyone out on here : p its also me to revise so a bunch! ask more questions if you dont understand ☺

hello from mrbilldoesmath!

answer:

(7 x + 6 y) (49 x^2 - 42 x y + 36 y^2)

discussion:

the trick is to recognize that 343 = 7^3 and 216 = 6^3 and, in general,

a^3 + b^3 =   (a + b) (a^2 - a b + b^2).

in our case

343x^3+216y^3   =   (7x)^3 + (6y)^3

so take a = 7x and b = 6y in the above equation to get the answer.

you,

mrb