An array of numbers is constructed beginning with the numbers $-1\qquad3\qquad1$in the top row. Each adjacent pair of numbers is summed to produce a number in the next row. Each row begins and ends with $-1$ and $1$, respectively.
$$\large{-1}\qquad\large{3}\qquad\large{1}$$
$$\large{-1}\qquad\large{2}\qquad\large{4}\qquad\large{1}$$
$$\large{-1}\qquad\large{1}\qquad\large{6}\qquad\large{5}\qquad\large{1}$$
If the process continues, one of the rows will sum to $12,288$. In that row, what is the third number from the left?
$\textbf{(A) } -29 \qquad\textbf{(B) } -21 \qquad\textbf{(C) } -14 \qquad\textbf{(D) } -8 \qquad\textbf{(E) } -3$
Difficulty: 48/100 — May encourage brute forcing, but overall quite feasible to solve.
Core Concepts: arithmetic & geometric sequences, Pascal's triangle
Challenges: Deriving the row sum formula; calculating the third term without resorting to brute force.
Techniques: Pattern recognition of the geometrically increasing row sums; inductive reasoning.
Error-prone Steps: Off-by-one error when finding the desired row.
Ideal Time:
Experienced: ≤ 1.5-2 min
Intermediate: ≤ 3-4 min
Beginner: ≤ 8 min