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Chapter 18: Problem 710

The arithmetic mean of 7 consecutive integers starting with a is \(m\) then the arithmetic mean of 11 consecutive integers starting with \(a+2\) is (a) \(2 \mathrm{a}\) (b) \(2 \mathrm{~m}\) (c) \(a+4\) (d) \(m+4\)

Short Answer

Expert verified
The arithmetic mean of the second set of 11 consecutive integers in terms of the mean of the first set is (d) \(m+4\).

Step by step solution

01

Find the mean of the first set of integers

To find the arithmetic mean of the 7 consecutive integers starting with a, we need to add all the integers and then divide the sum by 7. The integers are: a, a+1, a+2, a+3, a+4, a+5, and a+6. Mean of 7 integers, m = \(\frac{a+(a+1)+(a+2)+(a+3)+(a+4)+(a+5)+(a+6)}{7}\)
02

Simplify the mean for the first set

Now, we can simplify the mean expression by adding all the terms and dividing by 7: m = \(\frac{7a+21}{7}\) Divide both numerator and denominator by 7: m = \(a+3\) Now we have the arithmetic mean of the first set of consecutive integers in terms of a.
03

Find the mean of the second set of integers

Now we need to find the arithmetic mean of the 11 consecutive integers starting with a+2. The integers are: a+2, a+3, a+4, a+5, a+6, a+7, a+8, a+9, a+10, a+11, and a+12. Mean of 11 integers = \(\frac{(a+2)+(a+3)+(a+4)+(a+5)+(a+6)+(a+7)+(a+8)+(a+9)+(a+10)+(a+11)+(a+12)}{11}\)
04

Simplify the mean for the second set

Next, we simplify the expression by adding all the terms and dividing by 11: Mean of 11 integers = \(\frac{11a+77}{11}\) Divide both numerator and denominator by 11: Mean of 11 integers = \(a+7\) Now we have the arithmetic mean of the second set of consecutive integers in terms of a.
05

Compare the means and choose the correct option

We know the mean of the first set of 7 integers is m = \(a+3\), and the mean of the second set of 11 integers is \(a+7\). The question asks for the arithmetic mean of the second set in terms of m or a. Comparing the mean of the second set with the first set (m = \(a+3\)), we notice that the mean of the second set is 4 more than the first set's mean: Mean of 11 integers = m + 4 This matches option (d). So the correct answer is (d) \(m+4\).

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Most popular questions from this chapter

The median of a set of 7 distinct observations is \(10.5\) If each of the last 3 observation of the set is increased by 3 then the median of the new set \(=\) (a) in decreased by 2 (b) is two times the original median (c) remain the same as that of the original set (d) is increased by 2

The daily pocket expanses of 8 student are \(20,17,8,15,22\), \(9,10,14\) the median of the data is (a) \(14.5\) (b) 14 (c) 15 (d) \(15.5\)

A, B, C can hit the target with probability \((1 / 2),(1 / 3),(1 / 4)\) respectively. What is the probability that exactly two of them can hit the target? (a) \((1 / 2)\) (b) \((1 / 3)\) (c) \((1 / 4)\) (d) \((1 / 5)\)

Three identical dice are rolled. The probability that the same number will appear on each of them is (a) \((1 / 6)\) (b) \((1 / 36)\) (c) \((1 / 216)\) (d) \((1 / 18)\)

\(A, B\) and \(C\) can solve \(50 \%, 60 \%\) and \(70 \%\) of the sums from a book. If one sum from that book is given them to solve then probability that the sum will be solved is (a) \(0.94\) (b) \(0.06\) (c) \(0.47\) (d) None
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