Today's quiz is a very straightforward one. What is the sum of all of the consecutive whole numbers (integers) from 1 to 1,000, inclusive? [Hint: Manual addition will take you far too long, and introduces a high probability of error -- there might be a formula to help solve this one...]
Solutions to 12/23/08 Quiz:
What is the next number in each of the following sequences?
a) 1, 2, 4, 8, 16, 32, _____? The next number is 64. Each number can be found by doubling the one which precedes it. Alternately, each number in the series represents 2 to an exponential power. For example: 2 to the 0 power is 1; 2 to the 1st power is 2; 2 to the second power is 4, etc.
b) 1, 4, 27, 256, 3125, _____? The next number is 46,656. Each number in this series is a number to its own exponential power. For example: 1 to the first power is 1; 2 to the second power is 4; 3 to the third power is 9, etc.
c) 123, 234, 345, 456, 567, ______? The next number in this series is 678. Each of the numbers is generated by taking the preceding number, taking the middle integer, and using it to start a three-integer chain of ordinal numbers. Other than this if you add each of the three integers in each number, you will also find that it is 3 greater than the number which preceded it.
d) 10, 1011, 1011000, 10110001111, 101000111100000, ______? The next number is 101000111100000111111. This series is built on simply taking each number and adding successive integers to it (either zero or one, alternating) in an amount greater than the amount in the number which preceded it. For example, the first number is 10, the second is 1011 (adding 1 twice to the end of the number), the next number is 1011000 (adding 0 three times to the end of the number).
e) 1, 2, 6, 24, 120, _______? The next number is 720. The numbers in this series are each produced by taking the previous number and multiplying it by a number which is one greater than that which was used in arriving at the preceding number. For example: 1x2 =2; 2x3 =6; 6x4 = 24; 24x5 = 120, etc.
Solution to 12/24/08 Quiz:
Seven gentlemen are getting ready to leave a business meeting. They have all just met for the first time, and each of them will want to shake hands with each of the others. How many handshakes will be exchanged? (Remember: If two gents shake hands together, that only counts as one handshake. Also: no gentleman shakes hands with himself, unless he is praying, very cold, or addle-witted).
There is a formula for combinations which gives us the answer:
If n is the number of persons, and k is the amount of each combination, we divide n! by the product of (n-k)! x (k!) In this case, n=7 gents, k= a combination of 2, and ! means factorial (which means that number multiplied by each integer that precedes it. 4! would equal 4x3x2x1). In our case, this formula, with its blanks filled in would be 7!/(7-2)! x (2!), or 7!/5! x 2!, or (7 x 6)/2...which equals 21 handshakes in total.
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