Umar,
You can use differences to see how to generate the terms in the sequence.
sequence 
1 

7 

15 

26 
first differences 

5 

8 

11 

second differences 


3 

3 


If the second differences are all 3's then I can see how to generate the next term. The next second difference is 3 so the next first difference is 11 + 3 = 14 and thus the next term in the sequence is 14 + 26 = 40.
sequence 
1 

7 

15 

26 

40 
first differences 

5 

8 

11 

14 

second differences 


3 

3 

3 


And now you can generate the next term and so on but you asked for an expression for the nth term. To do this I need some notation. I am going to call the nth term of the sequence x_{n} and the nth first difference d_{n} . Then the table of differences around the nth term looks like
sequence 
x_{n2} 

x_{n1} 

x_{n} 
first differences 

d_{n1} 

d_{n} 

second differences 


3 


Thus
d_{n} = x_{n}  x_{n1}
d_{n1} = x_{n1}  x_{n2} , and
d_{n}  d_{n1} = 3
Substitute the values for d_{n} and d_{n1} from the first two equations into the third equation and solve for x_{n}. This will give you an expression for x_{n} in terms of x_{n1} and x_{n2}.
Penny
