Write the Recursive Formula

Question

Write a recursive formula for the sequence 1, 3, 6, 10... Is the sequence linear?

Answer 1

1,3,6,10,15,21,28,38,47,57

the difference between each term is 1 greater than the previous

since it is NOT a common difference, the sequence is not linear

f(1) = 1

f(2) = 3

f(n) = f(n-1) + f(n-1)-f(n-2)+1

= 2 * f(n-1) - f(n-2) + 1

So f(3) = 2 * f(2) - f(1) + 1 = 2 * 3 - 1 + 1 = 6

f(4) = 2 * f(3) - f(2) + 1 = 2 * 6 - 3 + 1 = 12 - 3 + 1 = 10

f(5) = 2 * f(4) - f(3) + 1 = 2 * 10 - 6 + 1 = 20 - 6 + 1 = 15