13 Rows Are Quite A Lot!

Relevant wiki: General Term Pattern Recognition

Notice that the last term on the $nth$ row is $n^2(*)$ ,so the last term on the $12th$ row is $144$ ,so the first term in the $13th$ is $145$ ,and the $2nd$ is $\boxed{146}$ .

$(*)$ proof:

After each down step,we add two columns.So the $nth$ row has $2n-1$ terms,as the first has one column.As the number of the term determines its value,the last term on the $nth$ row is $\sum _{ k=1 }^{ n }{ 2k-1 } =n^{ 2 }\star$ ,which is just the sum of how many terms there were.

Proof of $\star$ :

$\sum _{ k=1}^{ n }{ 2k-1 } =2\sum _{k=0}^{n}{k}-n=n(n+1)-n=n^2+n-n=\boxed{n^2}$

Let $a_n$ denote the first term in nth row and
$S_n$ be their sum.

$\begin{aligned}S_n=1+&2+5+\cdots+a_n\\S_n=~~~~~~&1+2+5+\cdots+a_n\end{aligned}$

Subtracting and rearranging for $a_n$ gives:-

$a_n=1+\underbrace{(1+3+5+\cdots+\text{(n-1) terms})}_{(n-1)^2}$

$\implies a_n=(n-1)^2+1$

Thus in general nth row contains $(n-1)^2+1,(n-1)^2+2,\cdots ,(n-1)^2+2n-1$

Hence 2nd term in 13th row is $(13-1)^2+2=\boxed{146}$ .

Central numbers are 1 3 7 13 therefore in nth row central number is n^2-n+1. In row 13 the central number is 169-13+1=157. The number of entries in each row are 1 3 5 7 so in the nth row there are 2n-1 entries. In row 13 that is 25. Therefore the row starts with 157-12=145 and so the second number is 146.

Observe that the first number of the $n^{\text{th}}$ row is given by $1$ + sum of first $n-1$ odd numbers.
So, the first term of the $13^{\text{th}}$ row is $1 + {(13 - 1)}^2 = 1 + 144 = 145$ . (Sum of first $n$ odd numbers is $n^2$ ).

So, the second term of this row is $145 + 1 = \color{#3D99F6}{\boxed{146}}$ .

Now, why does the first term increase like that?
It is because each term increases by the next odd number. So, the first term of the succeeding row would have 2 terms more than the current one (difference between two consecutive odd numbers is 2).

observe the pattern is square numbers

1 line = 1

2 lines = 4

3 lines = 9

so at the end of the 12th row we are at 144, so the second entry in row thirteen is is just a simple as add 2, giving 146.

Simple way is to work out 2 is added to every row. Add the first 12 odd digits then plus 2.

row 1's first number= 1 row2's first number= 2 row3's first number= 5 row4's first number= 10 . . . . row n's first number= n^2 - 2n + 2

Therefore row 13's first number= 169-26+2=145 Therefore row 13's second number= 145+1=146

Since Sn=n/2(2a+(n-1)d);

a=first element =1 ; d=difference of elements between two rows =2 ; n=no.of rows.

so here we have to find the 2nd element of 13th row. Therefore we will calculate till n=12;

so S12=12/2 ( 2+(12-1)*2)

S12=6*24=144;

so 2nd element from 144 i.e 146.

There are two more terms on every successive row than the previous and the first row contains one term. This allows us to deduce that there are $2n+1$ terms on every $(n+1)^\text{th}$ row. The square number that succeeds $a^2$ can be represented as $(a+1)^2$ which equates to $a^2+2a+1$ . Therefore, the last term in every $n^\text{th}$ row $=n^2$ . Then, the last term on the $12^\text{th}$ row is 144. Add 2 to get the 2nd term on the $13^\text{th}$ row $=\boxed{146}$ .