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Difficulty: MediumConsecutive Integers and Number Sets

Set SS consists of nn consecutive odd integers, ordered from least to greatest. The arithmetic mean of the integers in set SS is 4242. If the sum of the smallest integer and the largest integer in set SS divided by nn equals 77, what is the largest integer in set SS?

  1. A
    31
  2. B
    48
  3. 53Answer
  4. D
    55
  5. E
    61

Answer

The largest integer in set SS is 53.
For an evenly spaced set of numbers, the arithmetic mean is the average of the first and last terms. Therefore, (a1+an)/2=42(a_1 + a_n) / 2 = 42, which means a1+an=84a_1 + a_n = 84. Using the given condition (a1+an)/n=7(a_1 + a_n) / n = 7, we substitute 84/n=784 / n = 7 to find n=12n = 12. Since consecutive odd integers increase by 2, the difference between the 12th term and the 1st term is (12−1)×2=22(12 - 1) \times 2 = 22. Solving the system an+a1=84a_n + a_1 = 84 and an−a1=22a_n - a_1 = 22 yields 2an=1062a_n = 106, so the largest integer ana_n is 53.

Step-by-Step Solution

1
Express the sum of the smallest and largest integers using the mean.
Sum of smallest and largest integers =84= 84
In any set of consecutive integers (an arithmetic progression), the mean is equal to the average of the first term a1a_1 and the last term ana_n: a1+an2=42  ⟹  a1+an=84\frac{a_1 + a_n}{2} = 42 \implies a_1 + a_n = 84.
2
Determine the number of elements nn in set SS.
n=12n = 12
The stem specifies that a1+ann=7\frac{a_1 + a_n}{n} = 7. Substituting a1+an=84a_1 + a_n = 84 gives 84n=7  ⟹  n=12\frac{84}{n} = 7 \implies n = 12.
3
Calculate the difference between the largest and smallest integers.
an−a1=22a_n - a_1 = 22
In a sequence of nn consecutive odd integers, the common difference between terms is 22. The number of gaps between the 1st term and the nn-th term is n−1=11n - 1 = 11. Thus, an−a1=11×2=22a_n - a_1 = 11 \times 2 = 22.
4
Solve for the largest integer ana_n.
an=53a_n = 53
Adding the equations an+a1=84a_n + a_1 = 84 and an−a1=22a_n - a_1 = 22 yields 2an=106  ⟹  an=532a_n = 106 \implies a_n = 53.

Key Concept

Properties of consecutive odd integers and evenly spaced sets (mean-median equivalence and indexing)
Estimated Time:1m 30s
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Topics

Consecutive Integers and Number Sets

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