6th grade Math
posted by Jordan on .
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
51=4
145=9
3015=16
94=5
169=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