Question of the Day! October 2 ~ October 8

The following questions can be answered in the next few days!

Hi, everyone! Sorry I am late. But here are some more questions!

Questions

1. Suppose that a and b are real numbers.

If $\frac{a}{2} + \frac{b}{2^2} + \frac{a}{2^3} + \frac{b}{2^4} + \cdots + \frac{a}{2^{(2n-1)}} + \frac{b}{2^{(2n)}} + \cdots = 3$

Then, what is the value of $a+b$?[40 points]

2. Suppose that a and b (again) are positive integers this time. If b² = 9a, and if a + 2b > 280, what is the smallest possible number for a? [35 points] No lucky prime hint today (again). Though, you can still guess it! It is a number between 200 ~ 400.

3. You received 4 poker cards, assuming it is a standard 52-card deck, what are the chances to get exactly 4 aces? [35 points]

Answers for last time

For number 1: Same as last time.

For number 2: Correct! Eventually by comparing both equations, it will eventually lead to x = -2. Then, y can be easily found that y = 1/16.

Lucky Prime Number: 37 is no longer correct. Try a different one!

Leaderboards

1. Blueeighthnote: 13843 points (2 achvmt pkgs) - One more to get the next lucky prime badge!
While being the host, the host cannot earn any points during the period.

2. Minipop56: 12206 points - Four more to get the next lucky prime badge!

3. TimBluesWin: 755 points (1 achvmt pkg)

4. Dalek-James: 760 points

5. Imamadmad: 745 points (1 achvmt pkg)

6. Julianthewiki: 525 points (1 achvmt pkg) - RETIRED

7. 3primetime3: 493 points ("Out of Ideas" Badge)

8. Lefty7788: 300 points - One more to get the next lucky prime badge!

9. Wikia Contributor 69.235...: 230 points (1 achvmt pkg)

10. Supermario3459: 145 points (1 achvmt pkg)

11. ZombieBird4000: 50 points (1 achvmt pkg)

12. Wildoneshelper: 40 points (1 achvmt pkg)

13. Emmaelise401: 50 points

14. CalzoneManiac: 30 points

15. 21qmcgagin: 20 points

16. Kyodaisuu: 21 points

17. Googleaarex: 10 points

Achievements

The Badges will still be continued further.

Skillz Badges!

Usually awarded for having skills to solving hard questions, etc.

1. Quick Thinker! (+10 points) - Awarded for posting an answer before I write "Please leave your answers here." FIRST FOUNDED BY BLUEEIGHTHNOTE

2. RARE: Leaderboards Expert! (+300 points) - Awarded for being 1st place on the leaderboards for 30+ days. FIRST FOUNDED BY BLUEEIGHTHNOTE

3. RARE: More than a Millenium! (+100 points) - Awarded for being first in the leaderboards and scoring more than 1000 points from the runner-up. FIRST FOUNDED BY BLUEEIGHTHNOTE

4. Questions in a question (+20 points)? Awarded for finding three ambiguities in a single question. EARNED BY IMAMADMAD

Earth Badges

Most rare ones are found here.

1. RARE: Top Ten (+30 points) - Awarded for being the first top 10 people ONLY to participate on the question of the day! EARNED BY THE FIRST 10 PARTICIPATING PEOPLE

2. RARE: Cyclone Centurion! (+250 points) - Awarded for scoring at least 10000 points in the QOTD system! FIRST FOUNDED BY BLUEEIGHTHNOTE.

Sun Badges

All epic ones are found here.

1. EPIC: Lucky Primer (+100 points): Congratulations! You have guessed the lucky prime correctly. Get the Lucky Prime two more times to earn the next achievement! FIRST FOUNDED BY JULIANTHEWIKI.

2. EPIC: Very Lucky Primer (+200 points): Congratulations! You have guessed the lucky prime correctly two more times. Get the Lucky Prime three more times to earn the next achievement! FIRST FOUNDED BY MINIPOP56 AND BLUEEIGHTHNOTE.

3. REALLLLY LUCKY PRIMER (+300 points): Congratulations! You have guessed the lucky prime correctly three more times. Get the Lucky Prime four more times to earn the next achievement! FIRST FOUNDED BY MINIPOP56.

4. EPICEPIC: First of 2015! (+350 points): Congratulations for being the first to answer the questions in 2015 QOTD! FIRST FOUNDED BY DALEK-JAMES.

Galaxy Badges

Mysterious badges from a far-out universe. Each one found here is worth 10000+ points and/or 5+ achievement packages.

1. MYSTERY: ATLAS (+10000 points) - ??? FIRST FOUNDED BY BLUEEIGHTHNOTE.

2. FOUNDER: TIME KEEPER (+0 points) -- Being the host of QOTD for more than 1 year! Special thanks to your dedications! FIRST FOUNDED BY 3PRIMETIME3!

Uniqueness Badges

Unique things one can do in the answers section. ONLY ONE PERSON CAN EARN EACH BADGE.

1. Analyzer! (+10 points) - Awarded for drawing a diagram. FOUNDED BY IMAMADMAD

2. Inspiration! (+5 points) - A question was dedicated to you! FOUNDED BY BLUEEIGHTHNOTE

3. Out of Ideas. (+20 points) - Awarded for simply out of ideas. FOUNDED BY 3PRIMETIME3

Trustworthiness Badges

Awarded by how much I trust someone on this wiki. Please do not ask me about these badges.

1. Reliability (+100 points) - Trusted to write a question of a day. FIRST FOUNDED BY BLUEEIGHTHNOTE, AND IMAMADMAD

2. Scoreboard Tinkerer (+100 points) - Trusted to change the scoreboards. FIRST FOUNDED BY BLUEEIGHTHNOTE AND JULIANTHEWIKI

3. Prime Staff (+100 points) - Become a chat moderator, rollbacker, sysop, or bureaucrat on this wikia! FIRST FOUNDED BY BLUEEIGHTHNOTE, JULIANTHEWIKI, TIMBLUESWIN, IMAMADMAD, SUPERMARIO3459, MINIPOP56, AND DALEK-JAMES

Duration Badges

Yes! YOU KNOW IT! PARTICIPATION EVERY DAY!

1. 4-Leaf Clover! (+100 points) - Awarded for participating on the Question of the Day everyday during the month of March! FIRST FOUNDED BY MINIPOP56.

2. April Fools! (+100 points) - Awarded for participating on the Question of the Day everyday during the month of April! FIRST FOUNDED BY MINIPOP56.