Pattern Problem and Word Problem (doubling pennies, etc)

[geekily]

I've been staring at my homework for hours, and my boyfriend, who's much better at math than I am, has been able to help me with all the problems except for these 2:

1) Find the missing term in the pattern: 127, 863 12, 789 _____ 135 18

It's written exactly like that, with commas only after 127 and 12, so we tried figuring out if they were number sets that had something to do with each other, and tried adding up the numbers to see if there was any pattern, but there wasn't. The answer in the book is 1287, but I don't understand how they got that.

2) Would you rather work for a month (30 days) and get paid 1 million dollars or be paid 1 cent the first day, 2 cents the second day, 4 cents the third day, 8 cents the fourth day, and so on? Explain.

I know there's a way to do this other than 1 + 2 + 4 + 8 + 16, but I just don't see it. We've been working with the formula n (n + 1) / 2 to solve problems with sequential numbers, but I'm not sure how to apply that to this problem.

If you could help me with either one, I'd really appreciate it! Thank you so much!

 

[Denis]

Re: Pattern Problem and Word Problem

Find the missing term in the pattern: 127, 863 12, 789 _____ 135 18

It's written exactly like that, with commas only after 127 and 12, so we tried figuring out if they were number sets that had something to do with each other, and tried adding up the numbers to see if there was any pattern, but there wasn't. The answer in the book is 1287, but I don't understand how they got that.

Looks like it's gotta do with 1278: from 127,8 to 12,78 ; that's all I can see

Would you rather work for a month (30 days) and get paid 1 million dollars or be paid 1 cent the first day, 2 cents the second day, 4 cents the third day, 8 cents the fourth day, and so on? Explain.

That's an "oldie"; see it here:
http://mathforum.org/dr.math/faq/faq.do ... nnies.html

 

[geekily]

Re: Pattern Problem and Word Problem

Thanks, Denis! I really appreciate it!

 

[soroban]

Hello, geekily!

1) Find the missing term in the pattern: \(\displaystyle \:127,863\;\;12,789\;\;\fbox{\;\;?\;\;}\;\;135\;\;18\)

Once I figured out what those commas meant, I got it . . . it's a sneaky one!

The first number is \(\displaystyle 127,863\)
Add the last digit \(\displaystyle (3)\) to "the rest of the number": \(\displaystyle \:12786\,+\,3\:=\:12789\)

Do the same thing to \(\displaystyle 12789\)
Add the last digit \(\displaystyle (9)\) to the rest of the number: \(\displaystyle \:1278 + 9\:=\:\fbox{1287}\)

Keep going . . .
Add the last digit \(\displaystyle (7)\) to the rest of the number: \(\displaystyle \:128 + 7 \:=\:135\)

Add the last digit \(\displaystyle (5)\) to the rest of the number: \(\displaystyle \:13 + 5\:=\:18\)

See? . . . We're right!

2) Would you rather work for a month (30 days) and get paid 1 million dollars
or be paid 1¢ the first day, 2¢ the second day, 4¢ the third day, 8¢ the fourth day,
and so on? .Explain.

You are right: the second plan pays: \(\displaystyle S \:=\:1\,+\,2\,+\,4\,+\,8\,+\,\cdots\,+\,2^{29}\) cents.

This is a geometric series.
. . It has: first term \(\displaystyle a\,=\,1\), common ratio \(\displaystyle r\,=\,2\), and \(\displaystyle n\,=\,30\) terms.

The sum of a geometric series is: \(\displaystyle \:S \:=\:a\,\frac{1\,-\,r^n}{1\,-\,r}\)

So we have: \(\displaystyle S\:=\:1\cdot\frac{1\,-\,2^{30}}{1\,-\,2} \:=\:2^{30}\,-\,1\)


The second plan pays: \(\displaystyle \:2^{30}\,-\,1\:=\:1,073,741,823\) cents \(\displaystyle \:=\:\$10,737,418.23\)

It's a no-brainer . . . take the second deal!