CPhill

We can write

a1 + (a1 + 2d) + (a1 + 4d) + (a1 + 6d) + (a1 + 8d) + (a1 + 10d) =  6a1 + 30d = 0

(a1 + d) + (a1 + 3d) + (a1 + 5d) + (a1+ 7d)+ (a1 + 9d) + (a1 + 11d) = 6a1 + 36d = 0

So

6a1 + 30d  = 0   

6a1 + 36d  = 0

6a1 + 30d = 0

-6a1 - 36d = 0        add these

-6d = 0 

d = 0

So

6a1 + 30(0) = 0

6a1 = 0

a1  = 0

Simplify  as

-5s^2  -16s + 39

This is a parabola that opens downward.....there is  no minimum  value

https://web2.0calc.com/questions/sum_93

The psychology textbooks can be grouped together in one of  five positions....and  for each of these, they can be arranged 3!  = 6 ways

And for each of  the above arrangements the math textbooks can be  arranged in 4! = 24 ways

So

5 * 3! * 4!   =

5 * 6 * 24  =

720 ways

https://web2.0calc.com/questions/triangle_49107

Alpha  =  the measure of the interior angle of a 15-gon  (a pentadecagon) =

[ (n - 2) * 180 ]  / n  =    [ (15 - 2) * 180 ]  / 15  =  156° 

https://web2.0calc.com/questions/laser

II   only

To see why....let C inctrases by 1 and F increase by 1.8 = 9/5

C + 1  =  (5/9) ( F + 1.8 - 32)

C + 1 = (5/9) ( F + 9/5  - 32)         distribute 9/5

C + 1  = (5/9) ( F)  + (5/9)(9/5) - (5/9)(32)

C + 1 = (5/9) ( F - 32)  + 1            subtract 1 form both sides

C = (5/9) (F -32)       which is our original equation   

\($\log_{8a} b$ \)

Let this =  x

In exponential form

(8a)^x  =  b         take the log of both sides

log (8a)^x  =  log b      and we  can  write

x * log (8a)  = log b

x [ log 8 + log a ] = log b

x =     [ log b ] / [ log 8  + log a ]

Rearrange as

4x^2 + kx + 9   =  0

Will have non-real roors when the  discriminant  < 0

k^2  - 4 * 4 * 9 < 0

k^2  - 144 < 0

k^2  < 144

This will be true when k is on the interval ( -12 , 12)