Answer:
[tex]12942[/tex] is the sum of positive integers between [tex]1[/tex] (inclusive) and [tex]199[/tex] (inclusive) that are not multiples of [tex]4[/tex] and not multiples [tex]7[/tex].
Step-by-step explanation:
For an arithmetic series with:
there would be [tex]\displaystyle \left(\frac{a_n - a_1}{d} + 1\right)[/tex] terms, where as the sum would be [tex]\displaystyle \frac{1}{2}\, \displaystyle \underbrace{\left(\frac{a_n - a_1}{d} + 1\right)}_\text{number of terms}\, (a_1 + a_n)[/tex].
Positive integers between [tex]1[/tex] (inclusive) and [tex]199[/tex] (inclusive) include:
[tex]1,\, 2,\, \dots,\, 199[/tex].
The common difference of this arithmetic series is [tex]1[/tex]. There would be [tex](199 - 1) + 1 = 199[/tex] terms. The sum of these integers would thus be:
[tex]\begin{aligned}\frac{1}{2}\times ((199 - 1) + 1) \times (1 + 199) = 19900 \end{aligned}[/tex].
Similarly, positive integers between [tex]1[/tex] (inclusive) and [tex]199[/tex] (inclusive) that are multiples of [tex]4[/tex] include:
[tex]4,\, 8,\, \dots,\, 196[/tex].
The common difference of this arithmetic series is [tex]4[/tex]. There would be [tex](196 - 4) / 4 + 1 = 49[/tex] terms. The sum of these integers would thus be:
[tex]\begin{aligned}\frac{1}{2}\times 49 \times (4 + 196) = 4900 \end{aligned}[/tex]
Positive integers between [tex]1[/tex] (inclusive) and [tex]199[/tex] (inclusive) that are multiples of [tex]7[/tex] include:
[tex]7,\, 14,\, \dots,\, 196[/tex].
The common difference of this arithmetic series is [tex]7[/tex]. There would be [tex](196 - 7) / 7 + 1 = 28[/tex] terms. The sum of these integers would thus be:
[tex]\begin{aligned}\frac{1}{2}\times 28 \times (7 + 196) = 2842 \end{aligned}[/tex]
Positive integers between [tex]1[/tex] (inclusive) and [tex]199[/tex] (inclusive) that are multiples of [tex]28[/tex] (integers that are both multiples of [tex]4[/tex] and multiples of [tex]7[/tex]) include:
[tex]28,\, 56,\, \dots,\, 196[/tex].
The common difference of this arithmetic series is [tex]28[/tex]. There would be [tex](196 - 28) / 28 + 1 = 7[/tex] terms. The sum of these integers would thus be:
[tex]\begin{aligned}\frac{1}{2}\times 7 \times (28 + 196) = 784 \end{aligned}[/tex].
The requested sum will be equal to:
That is:
[tex]19900 - 4900 - 2842 + 784 = 12942[/tex].
