Here are the enumbers done ating values of the comsecutive whole numbers from 1 t0 3.
‡1
‡2
‡3
Consider selecting either the positive or the negative Cale of each integer. For example, we could select -1,
2. 3; or 1, 2, 3; or 1, 2, 3; etc. By using all such pos. sible selections, the following sums are possible: 6,
-4, -2, 0, 2, 4, and 6. For example, 6 is obtained by using all the negative values 1 + -2 + -3; the sum of 2 is obtained from 3 + -1 + 2; and the largest possible sum of 6 is obtained by adding all the positive values: 1 + 2 + 3. What sums can be obtained by using either the positive or the negative value of each integer for these sequences?
a.士1 +2+3+4
b.士1士2+3+4¢5
¢.‡1+2 +3+4+5+6
d. Look for patterns and state a general conjecture regarding the sums for
‡1+2 +3+4+5+6...
士n
where n is any whole number greater than 1.
a. By selecting the positive value for each integer, the sum would be 1 + 2 + 3 + 4 = 10. By selecting the negative value for each integer, the sum would be -1 + -2 + -3 + -4 = -10. Therefore, the possible sums for this sequence are 10 and -10.
b. By selecting the positive value for each integer, the sum would be 1 + 2 + 3 + 4 + 5 = 15. By selecting the negative value for each integer, the sum would be -1 + -2 + -3 + -4 + -5 = -15. Therefore, the possible sums for this sequence are 15 and -15.
c. By selecting the positive value for each integer, the sum would be 1 + 2 + 3 + 4 + 5 + 6 = 21. By selecting the negative value for each integer, the sum would be -1 + -2 + -3 + -4 + -5 + -6 = -21. Therefore, the possible sums for this sequence are 21 and -21.
d. By examining the patterns, we can see that for ‡1 + 2 + 3 + 4 + 5 + 6..., the sum will always be equal to the sum of the first n positive integers, but with the opposite sign depending on whether the negative or positive values are selected. The sum can be calculated using the formula (n(n+1))/2. Therefore, the possible sums for ‡1 + 2 + 3 + 4 + 5 + 6 +...士n where n is any whole number greater than 1 are (n(n+1))/2 and -(n(n+1))/2.