CPhill

.........

Here's one answer :

1^2 = 1

2^2  = 4

3^2  = 9

so

s^2   = n

Here's another using summation notation :

1 = 1

1 + 3  = 4

1 + 3 + 5  = 9

so

∞

Σ  [2s - 1]      =   1 + 3 + 5 + 7 + ..............

s=1

Mmmmmm.....the confusion over "formulas"  arises from the initial value of "s"

Actually, Complex"......if s = the stage number, then your formula  of 3s + 1  is incorrect, because at stage 1, s = 1 and we should have 4  blocks instead of just 1

I made a slight mistake in my formula  by using n instead of s  .......

I should have been....

3s - 2  = n,      assuming we can set s = 1   at the start

But....as to Melody's  assertion that it's the "same" formula re-written, we can see that it's not true because if it were, then

3s + 1  =  3s - 2   which implies that

1 = -2      !!!!

Melody's  "formula"  calculates the first three terms correctly......the 4th term would be 28 and the 5th term would be 85

If the intent of the sequence is to have a difference of 3 blocks at every stage, then her formula would not be correct.......if something else was intended, her formula might be correct and mine might be incorrect....

Who knows???

1  4  7  10  13

How about this pattern ????

1^2  - 0^2 + 0   =  1 + 0   = 1

2^2 - 1^2 + 1   =  3 + 1   = 4

3^2 - 2^2 + 2 =  5 + 2  = 7

4^2 - 3^2 + 3 = 7 + 3  = 10

5^2 - 4^2 + 4 = 9 + 4  = 13

6^2 - 5^2 + 5  =  11 + 5  = 16

=

[ (n)^2 - (n-1)^2] + (n - 1)  =  3n - 2

cot^-1( 2 + sqrt(2) + sqrt(3) + sqrt(6) )  = pi/24

cot [ pi/24]  = 2 + sqrt(3) + sqrt(2) + sqrt(6)

cot [pi/24]  =

cot [(1/2)pi/12]] =

1/ tan [(1/2)pi/12)] =

1/ [sqrt ( 1 - cos(pi/12))/ (1 + cos(pi/12)]  =  

sqrt( 1 + cos(pi/12)) / sqrt( 1 - cos(pi/12) ) =  [multiply top/bottom  by sqrt( 1 + cos(pi/12)) ]

[1 + cos (pi/12)] / sqrt( 1 - cos^2(pi/12)) =

[1 + cos(pi/12)] / sqrt( sin^2(pi/12)) =

[1 + cos(pi/12) ] / sin(pi/12)  =

1/sin(pi/12) + cos(pi/12) / sin(pi/12)  =

csc[pi/12] + cot[pi/12] =

1/ sin [ pi/12] + 1 / [tan [pi/12]  =

1 / [ sin[ pi/4 - pi/6]] + 1 / [ tan[ pi/4 - pi/6] ]

1/ [ sin(pi/4)cos(pi/6) - sin(pi/6)cos(pi/4)]  +  1/ ( [tan(pi/4] - tan[pi/6]] / [1 + tan(pi/4)tan(pi/6)] )  =

1 / [ sqrt(2)sqrt(3)/4] - (sqrt(2)/4) ] + [ 1 + 1/sqrt(3)]/ [1 - 1/sqrt(3)]  =

4 / [ sqrt(6) - sqrt(2)]  +  [ (sqrt(3) + 1)/ sqrt(3))]   / ( [ sqrt(3) - 1]/ sqrt(3) )

4 [ sqrt(6) + sqrt(2)]/ 4  +  [ sqrt(3) + 1] / [sqrt(3) - 1]  =  [ multiply the second term by [sqrt(3) + 1] on top/bottom ]

sqrt(6) + sqrt(2)  + [sqrt(3) + 1]^2 / 2  =

sqrt(6) + sqrt(2) + [3 + 2sqrt(3) + 1] / 2  =

sqrt(6) + sqrt(2) + [ 4 + 2 sqrt(3)] / 2  =

sqrt(6) + sqrt(2) + 2 + sqrt(3)  =

2 + sqrt(3) + sqrt(2) + sqrt(6)    ..........and the left side  =  the right side

l -5  + 4 l  =

l  -1   l   =

1

72 =

9 * 8  =

3^2 * 2^3

28.92% of 209252    =

0.2892 * 209252  =  60515.6784

2а(а-1)+3(а-1) =  (a - 1) (2a + 3)

4х-4у+ах-ау =     4(x - y) + a (x - y)   =    (4 + a) (x - y)

2х^2-ху =            x (2x  - y )

-4^2  = -16

So

-4^2+(-4^2)  =

-16 + (-16)  

-32