I have a word problem with a pattern. I just can't figure it out. - On the first night, Kim's mom gave her a nickel, on the second, she gave her five nickels, on the third, she gave her fourteen nickels and on the fourth night she gave her thirty nickels. Her mother asked Kim to figure out what she would receive on the tenth night. - I know this must be easy, but I just don't see it.
night 1=1
night 2=5
night 3=14
night 4=30
difference from 1 and 2 = 4
difference from 2 and 3 = 9
difference from 3 and 4 = 16
Do you see anything special in the differences?
firs form two columsn
1 1
2 5 4
3 14 9
4 30 16
if you take the difference between the number of nickels for consecutive days you get the sequence 4,9,16,...
that is, day2 nickels - day1 nickels = 4
day3 nickels - day2 nickels = 9
day4 nickels - day3 nickels = 16
notice that these are perfect squares, so we could continue that difference column as 4,9,16,25,36,49 ....
and build up our chart by working backwards.
the next row would be
5 55 25 and then
6 91 36 ... continue to row 10
there are more sophisticated ways to do this, but not suitable for grade 6 math
(I noticed that the third difference column gives me a constant of 2, suggesting that a cubic expression can be found for 1,5,14,30,...
but that would be beyond grade 6 math)
first night=1 nickle
second night=5 nickles
third night=14 nickles
fourth night=30 nickles
5-1=4
14-5=9
30-15=16
9-4=5
16-9=7
there for every day the mom gives the child two more nickles plus what she got the day before, so the next night the child should receive 54 nickles. I'm prety sure.
no
To solve this pattern, let's analyze the given information step by step:
On the first night, Kim received a nickel.
On the second night, she received five nickels, which is equivalent to 5 * 5 cents = 25 cents.
On the third night, she received fourteen nickels, which is equivalent to 14 * 5 cents = 70 cents.
On the fourth night, she received thirty nickels, which is equivalent to 30 * 5 cents = 150 cents, or $1.50.
Now, let's try to find the pattern in the given information. Notice that each night, the number of nickels Kim receives is increasing by a certain amount:
From night 1 to night 2, the number of nickels increased by 4 (from 1 to 5).
From night 2 to night 3, the number of nickels increased by 9 (from 5 to 14).
From night 3 to night 4, the number of nickels increased by 16 (from 14 to 30).
Now, we can see that there is a pattern in which the increase in the number of nickels is going up by the squares of consecutive natural numbers (4 = 2^2, 9 = 3^2, 16 = 4^2). This is known as a quadratic sequence.
To find out how many nickels Kim would receive on the 10th night, we can continue the pattern. From night 4 to night 5, the increase would be 25 (5^2). From night 5 to night 6, the increase would be 36 (6^2).
Using this pattern, we can calculate the number of nickels Kim would receive on the 10th night:
Night 4: 30 nickels
Night 5: 30 + 5^2 = 55 nickels
Night 6: 55 + 6^2 = 91 nickels
Night 7: 91 + 7^2 = 140 nickels
Night 8: 140 + 8^2 = 204 nickels
Night 9: 204 + 9^2 = 285 nickels
Night 10: 285 + 10^2 = 385 nickels
Therefore, Kim would receive 385 nickels on the tenth night.