2020 AMC 10A Problem 8

Name: 2020 AMC 10A Problem 8 — Video Solution
Uploaded: 2022-11-05T14:22:22-07:00
Duration: 2 h 28 min 50 s
Description: Po-Shen Loh presents the video solution to 2020 AMC 10A Problem 8, from "2020 AMC 10A LIVE Solve".

Question

What is the value of

\begin{aligned} &1+2+3-4+5+6+7-8\\ &\quad+\cdots\\ &\quad+197+198+199-200? \end{aligned}

9{,}800

9{,}900

10{,}000

10{,}100

10{,}200

Answer: B

Concepts:summation pairing and grouping

Difficulty rating: 1060

Small hint:

Group the expression into blocks of four terms

Big hint:

Find a formula for the

j

th block and sum over

50

blocks

Video solution:

Click to load, then click again to play

Written solution:

Group the terms in blocks of four:

(1+2+3-4)

+(5+6+7-8)

+\cdots

+(197+198+199-200)

. The

j

th block is

(4j-3)+(4j-2)

+(4j-1)-4j=8j-6

.

There are

50

blocks, so the sum is

\sum_{j=1}^{50}(8j-6)=8\cdot\dfrac{50\cdot51}{2}

-6\cdot50=9900

. Thus, B is the correct answer.