Alan

Thanks David, I obviously misread it as Carbon. Shows how important it is to use the formatting options to ensure clarity (or perhaps I should just get stronger spectacles!).

Circular measure sometimes refers to an angle measured in radians; but it's difficult to tell if this is what you are looking for without knowing more about the context of the question.  

Half-life

$$The radioactive decay equation is $N = N_0e^{-\lambda t}$ where $N$ is number of particles at time $t$, $N_0$ is the number of particles at time zero, and $\lambda$ is the decay constant. The half-life, $t_\frac{1}{2}$ is the time taken to get from $N_0$ to $\frac{1}{2}N_0$ and is related to the decay constant by $$t_\frac{1}{2}=\frac{ln(2)}{\lambda}$$

Hmm! It's a moot point which is the simpler of your two equations Melody.  Here's what the computer algebra system built into Mathcad thinks:

It should be noted that there isn't a mathematical definition of "simplify" or "simpler" and computer algebra systems often produce "simplified" expressions that don't match our human expectations.  Also, people often differ on what is simpler! 

Practise!  

Avogadro's number N_A = 6.0221415.10²³ molecules per mole

One mole of C₁₂ is 12 grams.

This means 6.0221415.10²³ molecules of C₁₂ is 12 grams.

So 5*10²⁶ molecules of C₁₂ is (12*5*10²⁶)/(6.0221415*10²³⁾ grams

$${\frac{\left({\mathtt{12}}{\mathtt{\,\times\,}}{\mathtt{5}}{\mathtt{\,\times\,}}{{\mathtt{10}}}^{{\mathtt{26}}}\right)}{\left({\mathtt{6.022\: \!141\: \!5}}{\mathtt{\,\times\,}}{{\mathtt{10}}}^{{\mathtt{23}}}\right)}} = {\mathtt{9\,963.233\: \!178\: \!762\: \!073\: \!259\: \!2}}$$ grams

But see David's reply below.

Is the first line meant to be the x-values and the second line the y-values? (Actually, the values in each line are sequential, though not with a constant interval in the case of the first line.)

If so, I suggest you first use your method of differences on each line separately to find x in terms of n (the position number of the items : for n = 1, 2, 3, 4 etc    x = 1, 3, 6, 10, etc.

Then eliminate n from the two equations to get x in terms of y.

So, you should find that x = n(n+1)/2  and y = 25n -1.  Eliminate n from these two equations to get x in terms of y.

However, I might have misunderstood what you are asking!

I think better with positive values!!  My 'a' isn't meant to be a multiplier, I just used ax as a two letter name for -x.

However, I think your method is much clearer!

Let ax be -x, so ax must be positive, given that x is negative.  So we have l-ax-3l = 5 or l-(ax+3)l = 5

The term in brackets is positive, and the absolute value operator means that the minus sign vanishes, so we are left with ax+3 = 5, which means that ax = 2.  This in turn means that x = -2.  

Similarly, let ay be -y. Then we have l-ay+7l = 15 or l-(ay-7)l = 15 (and ay must be larger than 7), so ay-7 = 15 , so ay = 22.  This means y = -22.

Therefore x - y = -2 - (-22) = -2 + 22 = 20

Yes, it is miles per second.  Also it is  $$5555\frac{5}{9}$$  which is 5555.6 (to nearest tenth) rather than 555.5.