Michelle is making 8 identical necklaces by stringing beads together. To make one necklace, se will repeat the pattern of beads shown four times. How many purple and yellow beads will be needed to make all eight necklaces?

This is the pattern:
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p -y -pp -y -ppp -y -pppp -y
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p means purple
y means yellow

First, count the number of p and y in one pattern.

Then, multiply these respective numbers by 4 because you need 4 times these patterns for one necklace.

Then you multiply by 8 because you want 8 such necklaces.

One pattern :
y=4, p=10

One necklace :
y = 4 x 4 = 16, p = 10 x 4 =40

8 necklaces :
y = 16 x 8 = 128, p = 40 x 8 = 320

To determine the number of purple and yellow beads needed to make all eight necklaces, we need to analyze the pattern of beads in one necklace.

Looking at the pattern provided, we can see that the number of beads increases by one with each repetition. In other words, the first repetition has one purple bead (p), the second repetition has two purple beads (pp), the third repetition has three purple beads (ppp), and so on.

We can also observe that there is one yellow bead (y) after each set of purple beads. This means that after the first purple bead (p), there is one yellow bead (y), after the second set of purple beads (pp), there is another yellow bead (y), and so on.

To find the number of purple beads in each repetition, we sum up the numbers from 1 to 8. This can be calculated using the formula for the sum of an arithmetic series:

sum = (n/2) * (first term + last term),

where n is the number of terms and the first term is 1, and the last term is 8.

Using the formula, we can calculate the sum:
sum = (8/2) * (1 + 8) = 4 * 9 = 36.

Therefore, the there are a total of 36 purple beads in each repetition.

To find the number of yellow beads, we need to consider that each repetition has one yellow bead. Since there are 8 repetitions in total, there will be 8 yellow beads.

Therefore, Michelle will need 36 purple beads and 8 yellow beads to make all eight necklaces.