What is the next number in the sequence 10 11 13 16 and why?

Answer 1

The next number is 42 as each term is given by:

t{n} = (11n⁴ - 110n³ + 391n² - 556n + 384)/12

Then again, it could be 20 because each term is given by:

t{n} = (n² - n + 20)/2

which corresponds to the difference between two term being one more than the difference between the previous two terms:

t2 - t1 = 1

t3 - t2 = 2 = 1 + 1

t4 - t3 = 3 = 2 + 1

so the next difference is 3 + 1 = 4, so the next term t5 would be t4 + 4 = 16 + 2 = 20.

Also: t{5} = (5² - 5 + 20)/2 = (25 - 5 + 20)/2 = 40/2 = 20.

In fact, what number do you want to come next? Give me the number you want to be next and I could give you a polynomial (relating to the term number n) which gives the results {10, 11, 13, 16} for terms n = 1, 2, 3, 4 and the number you specify for n= 5.

The second answer is probably what your teacher wants, though the first answer is a perfectly good answer.