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Zorluk: OrtaConsecutive Integers and Number Sets

Set SS consists of nn consecutive even integers. The arithmetic mean of all the integers in set SS is 4545. If the difference between the largest integer and the smallest integer in set SS is 3434, what is the value of the largest integer in set SS?

Cevap: 62

Cevap

The largest integer in set SS is 6262.
In any set of evenly spaced numbers, such as consecutive even integers, the arithmetic mean is equal to the average of the smallest term and the largest term. Since the arithmetic mean is 45, the sum of the smallest term and the largest term must be 45×2=9045 \times 2 = 90. Combining this with the given fact that the difference between the largest term and the smallest term is 34 creates a system of equations: Smallest + Largest = 90 and Largest - Smallest = 34. Adding these two equations cancels out the smallest term, resulting in 2 * Largest = 124, which gives 62 for the largest integer.

Adım Adım Çözüm

1
Express the arithmetic mean of the set in terms of the smallest element (FF) and largest element (LL).
F+L=90F + L = 90
For any evenly spaced set of numbers, the arithmetic mean is equal to the average of the first and last elements: F+L2=45\frac{F + L}{2} = 45.
2
Set up the equation for the difference between the largest and smallest elements.
L−F=34L - F = 34
The question specifies that the largest integer exceeds the smallest integer by 34.
3
Solve the system of two linear equations for LL.
L=62L = 62
Adding (F+L)+(L−F)=90+34(F + L) + (L - F) = 90 + 34 eliminates FF, leaving 2L=1242L = 124, which yields L=62L = 62.

Anahtar Kavram

Equivalence of arithmetic mean to the average of the first and last terms in an evenly spaced set
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Konular

Consecutive Integers and Number Sets

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