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, above, 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.
s3.amazonaws.com/tn8prev-prd-cn-tn8general/items/2/40f80328-e0b5-475f-92f3-8df5370988ba/assets/305-4909179-1535439182719.PNG
The figure shows 9 cards. One card is placed in the middle. 2 cards are placed above it, 2 cards are placed below, 2 cards are placed to the right and 2 cards are placed to the left.
If Rachel continues the same pattern, how many cards will be on the table after Rachel completes Step 4?
Write the first 6 terms of the sequence that represents the total number of cards on the table after each step.
Describe the pattern in the sequence.
To determine the number of cards that will be on the table after Rachel completes Step 4, we can look for a pattern in the sequence and then apply it.
Let's start by observing the sequence and determining the number of cards at each step:
Step 1: 1 card
Step 2: 5 cards (1 card in the middle + 2 above + 2 below)
Step 3: 9 cards (5 cards + 2 to the left + 2 to the right)
Step 4: ? cards
To find the number of cards at Step 4, let's analyze the pattern in the sequence. Notice that at each step, the number of cards increases by 4 compared to the previous step.
Now let's write the first 6 terms of the sequence that represents the total number of cards on the table after each step:
Step 1: 1
Step 2: 5
Step 3: 9
Step 4: 13
Step 5: 17
Step 6: 21
From the pattern we observed, we can conclude that the nth term of the sequence is given by the formula 4n - 3.
So, applying this formula to find the number of cards at Step 4:
4(4) - 3 = 16 - 3 = 13 cards
The number of cards on the table after Step 4 is 13.
In summary, the first 6 terms of the sequence are 1, 5, 9, 13, 17, 21. The pattern in the sequence is that each term is 4 more than the previous term (4n - 3).
geez.
Get out some cards and do a few steps. That will give you the sequence.