I need help with my pre algebra homework!

Question

I can not figure this out. A generous millionaire had an unusual plan for givng away money. Beginning on his birthday, he would give away $1 the first day, $3 the second day, $5 dollars the third day and so on. What is the TOTAL amount of money given away in 100 days?

Answer 1

Basically you want the sum of the first 100 odd numbers. Day 1 is the first odd number, day 2 is the second odd number, day 3 is the third odd number and so on. Seeing that this is pre-algebra you may not know the formula for finding the sum of a certain number of consecutive odd numbers. If you do I'll do the problem using the formula as well as using simple arithmetic.

1+3+5+7+9+...+191+193+195+197+199=? If you take the numbers from 1 to 200 you have 100 even numbers and 100 odd numbers, that's why 199 is the 100th odd number.

Here are two solutions using arithmetic...

1+199=200

3+197=200

5+195=200

7+193=200

9+191=200

and so on...

How many groups of 200 do you have?

There are 100 numbers and they are grouped by twos, so there are 50 groups of 200.

50*200=10,000

The answer is $10,000 given away in 100 days.

Here's another solution...

the sum of the first 2 odd numbers is 1+3=4=2²

the sum of the first 3 odd numbers is 1+3+5=9=3²

the sum of the first 4 odd numbers is 1+3+5+7=16=4²

Do you see the pattern ?

The sum of the first n odd numbers is n².

The sum of the first 100 odd numbers is 100²=100*100=10,000.

Using the formula...

S₍₁₀₀₎=(n/2)(2a+[n-1]*d) where n=100, a=1 (first term), and d=2 (the common difference between terms)

       =(100/2)(2*1+[100-1]*2)

       =50(2+99*2)

       =50(2+198)

       =(50)(200)

       =10,000