## Saturday, June 16, 2012

### Mathcounts strategies: Some sums

The following sequences are all arithmetic sequences and for any arithmetic sequences, the sum is
always average times the terms (how many numbers).

To find the average, you can use

a. sum divided by how many numbers.
b. average of the first and the last term, the second first and the second last term, or the third first and the third last term, etc...

Sum of the first consecutive natural numbers:

1 + 2 + 3 + 4 + 5 + ... + ( n -1 ) + n = $$\dfrac {n\left( n+1\right) } {2}$$

Examples :

#1: 1 + 2 + 3 + ... + 100 = $$\dfrac {100\left ( 101\right) } {2}=5050$$
#2: 1 + 2 + 3 + ... + 27 = $$\dfrac {27\left( 28\right) } {2}=378$$
#3: 4 + 8 + 12 + ... + 80 = 4 (1 + 2 + 3 + ... + 20) = $$\dfrac {4\times 20\times 21} {2}=840$$

Sum of the first consecutive natural even numbers: Proof without words

Have you noticed in this sequence, every number is double the numbers in the first example,
so you don't need to divide by 2.

$$2+4+6+\ldots +2n=n\left( n+1\right)$$

Examples:

#1: 2 + 4 + 6 + ... 100 = 50 * 51 = 2550
#2: 2 + 4 + 6 + 8 + 10 + 12 = 6 x 7 = 42
In this case, you can also find the midpoint, which is 7 and then 7 x 6 = 42
#3: 2 + 4 + 6 + ...420 = 210 * 211 = 44,310

Sum of the first consecutive odd numbers: Proof without words

In this special case, the mean is the same as how many numbers.
So it's easier if you find the mean by averaging the first and the last term and then square the mean.

1 + 3 + 5 + ... + ( 2n -1) = $$n^{2}$$

Examples:

#1: 1 + 3 + 5 + ... + 39 = $$\left[ \dfrac {\left( 39+1\right) } {2}\right] ^{2}$$ = (20^{2}\)
#2: 1 + 3 + 5 + ... + 89 = $$\left[ \dfrac {89+1} {2}\right] ^{2}=45^{2}=2025$$
#3: 1 + 3 + 5 + ... + 221 = $$\left[ \dfrac {221+1} {2}\right] ^{2}= 111^{2}=12321$$

Applicable problems:

#1: What is the sum of the first 40 consecutive positive integers?
#2: What is the sum of the first 40 consecutive positive even integers?
#3: What is the sum of the first 40 consecutive positive odd integers?
#4: How many times does a 12-hour clock strikes in one day if it strikes once on one o'clock, twice on two o'clock, etc...?
#5: At a game show,you win $100 for the first correct answer and$200 for the second correct answer, etc. How much do you win if you answer 8 questions in a row correctly?

#1: 820
#2: 1640
#3: 1600
#4: 156
#5: 100 + 200 + 300 ... + 800 = 100 (1 + 2 + 3 + ...8) = 100* $$\dfrac {8\times 9} {2}$$ =  3600