Question

The expressions $A = 1 \times 2$ $+ 3 \times 4$ $+ 5 \times 6$ $+ \cdots$ $+ 37 \times 38 + 39$ and $B = 1$ $+ 2 \times 3$ $+ 4 \times 5$ $+ \cdots$ $+ 36 \times 37$ $+ 38 \times 39$ are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers $A$ and $B.$

Answer

Answer: 722

Concepts:summation arithmetic sequence pairing and grouping

Difficulty rating: 1890

Small Hint:

Compute $B - A$ by pairing each product in $B$ with the product in $A$ that shares its even factor

Big Hint:

Each pair contributes $(2k+1)(2k)$ $- (2k-1)(2k) = 4k,$ and the leftover terms are $1$ and $-39$

Solution:

Subtract term by term: $\begin{aligned} &B - A = (1 - 39) \\ &\quad {}+ (2 \times 3 - 1 \times 2) \\ &\quad {}+ (4 \times 5 - 3 \times 4) + \cdots \\ &\quad {}+ (38 \times 39 - 37 \times 38). \end{aligned}$ Each parenthesized difference has the form $(2k+1)(2k)$ $- (2k-1)(2k) = 4k$ for $k = 1, 2, \ldots, 19.$

Therefore $\begin{aligned} B - A &= -38 \\ &\quad {}+ 4(1 + 2 + \cdots + 19) \\ &= -38 + 4 \cdot 190 = 722. \end{aligned}$

2015 AIME I Problem 1

Answer: 722

Difficulty rating: 1890

Small Hint:

Big Hint:

Solution:

Problem 1 in Other Years