Gino just opened a pizza restaurant. He had 1 customer the first day, 3 the second day, 6 the third day, and 10 the fourth day.
a. If this pattern continues, how many customers should gino expect on the seventh day? Day 10? Show the data table (for rate of change)
b. Does the number of customers represent a geometric sequence, an arithmetic sequence, or neither?
c. find an expression to represent the number of customers on day "n."
d. if this pattern continues, on what day will Gino first serve over 200 customers?
a. To find the pattern and predict the number of customers on the seventh day and the tenth day, we can analyze the rate of change between each day. Let's create a table to represent the data:
Day 1: 1 customer
Day 2: 3 customers (increase of 2 customers from the previous day)
Day 3: 6 customers (increase of 3 customers from the previous day)
Day 4: 10 customers (increase of 4 customers from the previous day)
Now, let's continue the pattern:
Day 5: Increase of 5 customers from the previous day (10 + 5 = 15)
Day 6: Increase of 6 customers from the previous day (15 + 6 = 21)
Day 7: Increase of 7 customers from the previous day (21 + 7 = 28)
So, Gino should expect 28 customers on the seventh day.
Day 8: Increase of 8 customers from the previous day (28 + 8 = 36)
Day 9: Increase of 9 customers from the previous day (36 + 9 = 45)
Day 10: Increase of 10 customers from the previous day (45 + 10 = 55)
Therefore, Gino should expect 28 customers on the seventh day and 55 customers on the tenth day.
b. To determine if the number of customers represents a geometric sequence, an arithmetic sequence, or neither, we need to check the differences between consecutive terms:
1 -> 3: Increase of 2
3 -> 6: Increase of 3
6 -> 10: Increase of 4
The differences between consecutive terms (2, 3, 4) do not form a constant pattern, so it is not an arithmetic sequence. However, if we look closely, we can observe that the differences themselves form an arithmetic sequence (increase by 1 each time: 2, 3, 4). This indicates that the sequence is a quadratic sequence, rather than geometric or arithmetic.
c. To find an expression to represent the number of customers on day "n," we need to find a pattern in the number of customers.
If we observe closely, we can see that the number of customers on each day forms a quadratic pattern. We can express it as a quadratic function represented by a variable "n," where "n" represents the day.
The pattern is as follows:
Day 1: 1 customer (n = 1)
Day 2: 3 customers (n = 2)
Day 3: 6 customers (n = 3)
Day 4: 10 customers (n = 4)
...
We can see that the number of customers on each day is the sum of the first "n" natural numbers. This sum can be represented by the formula: n(n + 1)/2.
Therefore, the expression to represent the number of customers on day "n" is: n(n + 1)/2.
d. To find on what day Gino will first serve over 200 customers, we need to solve the equation: n(n + 1)/2 > 200.
Simplifying the equation, we get: n^2 + n > 400.
By trial and error, we can find that when n = 20, the equation will be true. Therefore, Gino will first serve over 200 customers on the 20th day.
your numbers are
1
3
6
10
..
suppose we double those and notice their factors:
2 = 1x2
6 = 2x3
12 = 3x4
20 = 4x5
mmmmhhhh?
can you do anything with those hints?