In a group of five families, every family is expected to have a certain number…
2025
In a group of five families, every family is expected to have a certain number of children, such that the number of children forms an arithmetic progression with a common difference of one, starting with two children in the first family. Despite the objection of their parents, every child in a family has as many pets to look after as the number of children in the family. What is the total number of pets in the entire group of five families?
- A.
100
- B.
80
- C.
95
- D.
90
Attempted by 5 students.
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Correct answer: D
When each child in a family has as many pets as there are children in that family, a family with n children ends up with n × n = n2 pets in total. The total pets across several families is therefore the sum of the squares of the individual family sizes.
The five family sizes form an arithmetic progression starting at 2 with common difference 1, so they are 2, 3, 4, 5, and 6.
Each family's total pets equal the square of its own size: 22 for the first family, 32 for the second, 42 for the third, 52 for the fourth, and 62 for the fifth.
Adding the five squares (22 + 32 + 42 + 52 + 62) gives the total pets for the entire group.
Evaluating each term and summing: 4 + 9 + 16 + 25 + 36 = 90.
As a check, the sum of squares from 12 through 62 is a standard result equal to 91; subtracting the extra 12 for the family size of one child that is absent from this progression leaves 91 − 1 = 90, matching the direct computation.
So the total number of pets in the entire group of five families is 90.