Question

A sequence has the following pattern 3/2,4,15/2,12,35/2,24.Starting n=1 find the general term for the sequence?

Answer 1

I noticed that the highest denominator is 2, so let's just double every term ( can't forget to divide by 2 at the end)
so we have:
3 8 15 24 35 48 ....
the first differences are
5 7 9 11 13 ...
and the 2nd differences are
2 2 2 2 ...

since the 2nd difference is a constant it must be a quadratic of the form
an^2 + bn + c = k
when n = 1:
a+b+c = 3 -->#1
when n = 2
4a+2b+c= 8 -->#2
when n = 3
9a+3b+c = 15 -->#3

#3 - #2 :5a + b = 7
#2 - #1 : 3a+b = 5
Now subtract those last two:
2a = 2
a = 1

into 3a + b = 5
3+b = 5
b = 2

into a+b+c = 3
1+2+c=3
c = 0

so we have n^2 + 2n or n(n+2)

but remember we doubled our numbers, so let's back that up

<b>term(n) = n(n+2)/2</b>

check:
let n = 5
term(5) = 5(7)/2 = 35/2
YEAHHHHH