# The first term of the 100th line? [closed]

In the following arrangement what would be the first number in 100th line?

 1   2   3
4   5   6   7
8   9  10  11  12
13  14  15  16  17  18


$(100 \times 103 \div 2) - 1 = 5149$

Because this sequence is

a(n) = binomial(n+1, 2) + n - 1 = n(n + 3)/2 - 1.

Since OP perhaps questions the correctness of this answer ... Try it online!

• The question is about the first number in the 100th line, not the 100th number. Aug 20 '17 at 19:49
• Well yes - that IS the first number of the 100th line. The 100th number is obviously just 100.
– Rubio
Aug 20 '17 at 19:51

5149

Explanation

Note that the third number in each row is a triangle number. Specifically, where n is the row number and the first row has n=1, the third number is (n+1)(n+2)/2. So in row 100, the third number is (101)(102)/2 = 5151, so the first number of row 100 is 5151-2 = 5149

Note also that my formula is mathematically equivalent to Rubio's:

(n+1)(n+2)/2 - 2 = n(n+3)/2 - 1

However, my derivation of the formula, and therefore explanation of my derivation, is different from Rubio's.

• Since your answer is the same as Rubio's, and your derivation of it is equivalent to Rubio's, why did you post it? Aug 20 '17 at 21:58
• @GarethMcCaughan Because my explanation, and the way i found the equation, was very different. It just happens that the math is isomorphic. Aug 20 '17 at 22:01