Question 6

Explanation

It is given that the ages of Pak Asep's children form an arithmetic progression with the number of children $n = 6$. Let $U_n$ be the age of the $n$-th child.

From the problem information:

• The age of the $3$rd child is $18$ years: $U_3 = 18$.
• The age of the $5$th child is $16$ years: $U_5 = 16$.

In an arithmetic progression, the formula for the $n$-th term is $U_n = a + (n-1)b$. We can form the equations:

Subtract the first equation from the second to find the common difference ($b$):

Substitute $b = -1$ into the equation for $U_3$ to find the first term ($a$):

So, the age of the first child is $20$ years, and the age difference is $-1$ year. The ages of the six children are: $20, 19, 18, 17, 16, 15$.

The sum of all children's ages ($S_6$) can be calculated using the arithmetic series formula:

Thus, the sum of the ages of all Pak Asep's children is $105$ years.