Question

How do you get the general term for a sequence that has increasing common differences? For example: 1,2,4,7,11

Answer 1

notice that the differences are 1,2,3,4,...
but the differences of those differences are 1,1,1,...

Since we have constants in the second set of differences our sequence can be expressed as a 2nd degree expression
(had the third set of differences been constant, it would have been a cubic expression, etc)

so let the sequence be written as
t(n) = an^2 + bn + c

if n=1 ---> a + b + c = 1
if n=2 ---> 4a + 2b + c = 2
if n=3 ---> 9a + 3b + c = 4

subtract the first two ---> 3a + b = 1
subtract the last two ---> 5a + b = 2

now subtract those
2a = 1
a = 1/2
back in 3a+b=1
3/2 + b = 1
b = -1/2
back in a+b+c=1
1/2 - 1/2 + c = 1
c=1

<b>so t(n) = (1/2)n^2 - (1/2)n + 1
= (1/2)(n^2 - n + 2)</b>

checking for t(5)
= (1/2)(25-5+2)
=(1/2)(22) = 11

Answer 2

why do you let n=1, 2 and 3?