Recursion Warmup - An Odd Sum

• 13th odd number is: $\boldsymbol {2\times13-1=25}$ .

• Mean of these numbers is: $\boldsymbol {\frac {1+25}{2}=13}$ .

• Total numbers are $\boldsymbol {13}$ .

So the sum will be, $\boldsymbol {13\times13=}\color{#69047E} {\boxed {169}}$ .

One knows that each subshell of an atom adds successive odd numbers of orbitals, and that each orbitals holds two electrons. One also knows the rule that for an atom with valence shell of principal quantum number $n$ , there can be up to $2n^{2}$ valence electrons.

So one concludes that there are $n^{2}$ valence orbitals and thus the sum of a subshell's orbitals - and hence the sum of succesive odd numbers - is $n^{2}$ .

a=1; n=13; d=2

Sn=n/2{2a+(n-1)d} =13/2{2*1+(13-1)2} =169

Get the middle number and multiply by the amount of number. 13 x 13 = 69