Divide 100 loaves of bread among five men so the amount is arithmetic sequence and the sum of the largest is seven times the amount of the two smallest.
first equation:
a + a+d + a+2d + a+3d + a+4d = 100
5a + 10d = 100
a + 2d = 20 or a=20-2d
2nd equation
a+4d = 7(a + a+d)
a+ 4d = 14a + 7d
13a =-3d
13(20-2d) = -3d
260 - 26d = -3d
260 = 23d
d = 260/23 (or appr 11.3)
a = 20 - 520/23 = -60/23 (or appr. -2.6 ????)
Mathematically, these answers work for the above condition, but they don't make any real sense.
Check:
first gets -2.6
2nd gets 8.7
3rd gets 20
4th gets 31.3
5th gets 42.6 , they form an arithmetic sequence
(-2.6+8.7+20+31.3+42.6 = 100)
7 times the first two = 7(-2.6+8.7) = 42.7
the last one = 42.6 (close enough, error due to roundoff)
Something is wrong with your question!
To divide 100 loaves of bread among five men in an arithmetic sequence, we can start by assigning variables to represent the unknowns. Let's say the first term of the sequence is "a," and the common difference between each term is "d."
Using these variables, we can express the sequence of bread loaves for each man as follows:
1st man: a
2nd man: a + d
3rd man: a + 2d
4th man: a + 3d
5th man: a + 4d
Next, we're told that the sum of the largest term and the sum of the two smallest terms is equal to seven times the two smallest terms. In equation form, this can be written as:
(a + 4d) + (a + (a + d)) = 7(a + (a + d))
Simplifying the equation, we have:
2a + 5d = 14a + 7d
Rearranging and simplifying further, we get:
12d = 12a
d = a
Now that we know the common difference d is equal to the first term a, we can solve for a by substituting back into one of the previous equations. Let's use the equation for the first term:
a + 4a = 100
5a = 100
a = 20
Therefore, the first term a is 20, and the common difference d is also 20.
To find the number of bread loaves each man receives, we can substitute the values of a and d back into the sequence expressions:
1st man: a = 20 loaves
2nd man: a + d = 20 + 20 = 40 loaves
3rd man: a + 2d = 20 + 2(20) = 60 loaves
4th man: a + 3d = 20 + 3(20) = 80 loaves
5th man: a + 4d = 20 + 4(20) = 100 loaves
Therefore, the five men receive 20, 40, 60, 80, and 100 loaves of bread, respectively, resulting in an arithmetic sequence.