The arithmetic mean of a sequence of 15 integers is 78. The largest

Solution

• The arithmetic mean of a sequence of 15 integers is 78
• The largest value of the set is five times the smallest value of the set
• The mean of the set is equal to the median

• The maximum possible value for the range of the set

• We know that the range of a set = the largest value – the smallest value

• We know that the mean = the median = the 8th element of the sequence = 78
• So, to minimise the first element of the sequence, all the elements before the median element must be same. Let us assume it be “a”
• And, to maximise the last element of the sequence, all the elements after the median except the last element must be minimum = the median = 78
• The last element of the sequence will be = 5 * a

• Implies, sum of all 15 elements in the sequence = 78 * 15
• Thus, a + a + a + a + a + a + a + 78 + 78 + 78 + 78 + 78 + 78 + 78 + 5a = 78 * 15

o 12a = 78 * 15 – 78 * 7
o \(a = 78 * \frac{8}{12} = 52\)

• Therefore, range of the set = 5a – a = 4a = 4 * 52 = 208