CPhill

x /y = 3/2   →  y = (2/3)x

y/z = (2/3)x / z = 4/3  →  z = (3/4)(2/3)x  = (1/2)x     .....Therefore......

x / ( y + z)   =    x / [ (2/3)x + (1/2)x]   =  x (1)/ [x (2/3 + 1/2)]  = 1/ (2/3 + 1/2)  =  1/ [ 7/6]  = 6/7 

I assume we have   .....

a_n = 2n -3 

The first term =a₁ =  2(1)-3   = -1

And the 10th term = a₁₀ = 2(10) - 3 = 20 - 3 = 17    

And the sum of the first 10 terms =  10(a₁ + a₁₀)/2  =   10(-1 + 17)/2 = 5(16)  = 80

364 / 18   = divide numerator and denominator by 2  

182 / 9   =   20 + 2/9     ...and that's it !!!!

Yep....geometric.....the ratio is (2/3)

The series is given by:

(1/3)(2/3)^(n - 1)   for n ≥ 1

This series is not arithmetic, because there is no common difference ......the nth term - for n ≥ 2 - is given by:

14 + 7(n-1)^2  

ƒ(−1) if ƒ(x) = 3(x −1)² + 5?     we're just substituting "-1" into this for "x" and evaluating...so we have....

3 (-1-1)² + 5  =

3(-2)² + 5  =

3(4) + 5 =

12 + 5  =

17

ab + a + b = 524   →   b(1 + a)  =  [524 - a]    →   b =  [524 - a] / [ 1 + a]

bc + b + c =  146   →   [524 - a] / [ 1 + a]c +  [524 - a] / [ 1 + a] + c = 146   →

c (  [524 - a] / [ 1 + a] + 1 )  =  146 -  [524 - a] / [ 1 + a]    →

c (  [ 1 + a + 524 -  a] / [ 1 + a]  =  ( [ 146 + 146a - 524 + a] / [1 + a]     →

c (525)   =  (147a - 378)     →   c   =  (147a - 378)/ (525) 

cd + c + d   =  104    →    (147a - 378)/ (525)d  +  (147a - 378)/ (525) + d = 104     →

d( [ 147a - 378]/ (525)  + 1 )  =  104 -  (147a - 378)/ (525)    →

d ( 525 - 378 + 147a) / (525)  =  (54600 + 378  - 147a) / (525)    →

d (147 + 147a) =  (54978 - 147a)   →   147d (1 + a) =  (54978 - 147a)   →

d (1 + a)  = (374 - a)  →   d =   (374 - a) / (1 +a)

Therefore

abcd =  8! = 40320

a*  [524 - a] / [ 1 + a]  *  (147a - 378)/ (525) * (374 - a) / (1 +a)  = 40320 

a * (524 -a) (147a - 378)(374 - a)  = 40320 (525) (1 + a)^2   ...  simplify

147a^4 - 132384a^3 + 29147916a^2 -74078928a    =  40320 (525)(1 + a)^2

21a( 7a^3 - 6304a^2 + 1387996a - 3527568) = 40320 (525)(1 + a)^2

a ( 7a^3 - 6304a^2 + 1387996a - 3527568) = 40320(25)(1 + a)^2

7a^4 - 6304a^3 + 1387996a^2 - 3527568a = 1008000a^2 + 2016000a + 1008000

7a^4 - 6304a^3 + 379996a^2 - 5543568a - 1008000 = 0   factor

(a - 24) (7a^3 -6136a^2 +232732a + 42000)   = 0

a = 24  is the only integer solution

b =  [524 - 24] / [ 1 + 24]  = 500/25 = 20

c =  (147(24) - 378)/ (525) = 6

d =  (374 - 24) / (1 +24)  = 350 / 25  = 14

And  24 * 20 * 6 * 14 =  40320

And a - d = 24 - 14  = 10

(b+4)^2    =   (b + 4) (b + 4)   = b² + 4b + 4b + 16   =    b² + 8b + 16 

32m-12-24m+7 =   combine like terms

32m - 24m -12 + 7  =

8m -5