Respuesta :
Answer
The absolute difference between the sum of the multiples of two from 1 and 100, inclusive, and the sum of the multiples of three from 1 and 100, inclusive = 867
Explanation
To solve this, we need to first find the sum of multiples of 2 from 1 to 100 inclusive and find the sum of multiples of 3 from 1 to 100 inclusive. These set of numbers, form an arithmetic progression because all the terms have the same common difference
To do that, we need to note that the sum of an arithmetic progression is given as
[tex]\text{Sum = }\frac{n}{2}\lbrack2a+(n-1)d\rbrack[/tex]a = First term
n = number of terms
d = common difference
For the multiples of 2
a = 2
n = 50
d = 2
[tex]\begin{gathered} \text{Sum = }\frac{n}{2}\lbrack2a+(n-1)d\rbrack \\ \text{Sum = }\frac{50}{2}\lbrack2(2)+(49-1)(2)\rbrack \\ =\frac{50}{2}\lbrack4+(48\times2)\rbrack \\ =\frac{50}{2}\lbrack4+96\rbrack \\ =\frac{50}{2}(102) \\ =2550 \end{gathered}[/tex]For the multiples of 3
a = 3
n = 33
d = 3
[tex]\begin{gathered} \text{Sum = }\frac{n}{2}\lbrack2a+(n-1)d\rbrack \\ \text{Sum }=\frac{33}{2}\lbrack2(3)+(33-1)(3)\rbrack \\ =\frac{33}{2}\lbrack6+(32\times3)\rbrack \\ =\frac{33}{2}\lbrack6+96\rbrack \\ =\frac{33}{2}(102) \\ =1683 \end{gathered}[/tex]So,
The absolute difference = |(Sum of multiples of two from 1 to 100 incusive) - (Sum of multiples of three from 1 to 100 incusive)
= |2550 - 1683|
= 867
Hope this Helps!!!
