Rachel is placing playing cards on a table by following these steps.
Step 1: First she places one card in the middle of the table.
Step 2: Next she adds one more card directly to the left, to the right, and below the first card.
Step 3: Next she adds one more card directly to the left, to the right, above, and below the cards that were already on the table. The cards form the pattern shown.
a. If Rachel continues the same pattern, how many cards will be on the table after Rachel completes Step 4?
b. Write the first 6 terms of the sequence that represents the total number of cards on the table after each step.
c. Describe the pattern in the sequence.
a. Well, if Rachel continues the same pattern and completes Step 4, we need to do a bit of card counting. Let me grab my clown calculator... Ah, yes! By my calculations, there will be a whopping 13 cards on the table after Step 4. Quite the card collection, I must say!
b. Now, let's write down the first six terms of the sequence representing the total number of cards on the table after each step:
Step 1: 1 card
Step 2: 5 cards
Step 3: 9 cards
Step 4: 13 cards
Step 5: Well, that's the future, my friend! Let's not go there just yet.
c. So, if we take a closer look at the sequence, we can see a pattern emerging. Each step seems to increase the total number of cards by 4. It's like Rachel is a magician pulling cards out of thin air. Spectacular!
a. To find the number of cards on the table after Step 4, we need to continue the pattern described in the problem.
In Step 1, Rachel places 1 card.
In Step 2, Rachel adds 4 cards.
In Step 3, Rachel adds 8 cards.
From the pattern, we can see that the number of cards Rachel adds in each step doubles. So, in Step 4, Rachel will add 16 cards.
To find the total number of cards on the table after Step 4, we need to sum up the number of cards in each step.
Step 1: 1 card
Step 2: 1 + 4 = 5 cards
Step 3: 5 + 8 = 13 cards
Step 4: 13 + 16 = 29 cards
Therefore, after completing Step 4, there will be 29 cards on the table.
b. The first 6 terms of the sequence that represents the total number of cards on the table after each step are:
Step 1: 1
Step 2: 5
Step 3: 13
Step 4: 29
Step 5: 61
Step 6: 125
c. The pattern in the sequence is that each term is obtained by adding double the number of cards added in the previous step.
To answer these questions, let's break down the steps and look for a pattern.
Step 1: Rachel places one card in the middle of the table. The total number of cards on the table at this point is 1.
Step 2: Rachel adds one more card directly to the left, to the right, and below the first card. So, for step 2, Rachel adds 1 + 1 + 1 = 3 cards. The total number of cards on the table at this point is 1 + 3 = 4.
Step 3: Rachel adds one more card directly to the left, to the right, above, and below the cards that were already on the table. So, for step 3, Rachel adds 1 + 1 + 1 + 1 = 4 cards. The total number of cards on the table at this point is 4 + 4 = 8.
Now, let's analyze the pattern we have so far:
Step 1: 1 card.
Step 2: 4 cards.
Step 3: 8 cards.
From this pattern, we can observe that the number of cards doubles at each step.
a. If Rachel continues the same pattern, how many cards will be on the table after Rachel completes Step 4?
To find out, we apply the pattern to the current number of cards on the table. From step 3, we had 8 cards. If we double 8, we get 16. So, after completing step 4, there will be a total of 16 cards on the table.
b. Write the first 6 terms of the sequence that represents the total number of cards on the table after each step.
The sequence representing the total number of cards on the table after each step would be:
Step 1: 1
Step 2: 4
Step 3: 8
Step 4: 16
Step 5: 32
Step 6: 64
c. Describe the pattern in the sequence.
The pattern in the sequence is that each term is obtained by doubling the previous term. In other words, each step doubles the number of cards on the table compared to the previous step.
Cannot see pattern. You cannot copy and paste here.
Also, It would help if you proofread your questions before you posted them. In step 2, she did NOT put a card ABOVE the first card?