Question

5 11 21 35 53......



what is the next....?



and what is the formula?

Answer 1

This is an example of a quadratic sequence (a sequence which has $n^2$ term in it). The reason it is a quadratic sequence is that the difference between each two consecutive terms is changing. So now we need to take a look at the difference between the differences.

So the difference between the first and second term is 6. The difference between the second and third term is 10. The difference between the third and fourth term is 14. Therefore, we can conclude that the difference between the differences is 4.

In order to get the formula, first we need to divide the difference between the differences number by 2, which will be the coefficient of the $n^2$ term. Therefore, the coefficient is 4/2 = 2.

After this, we try to obtain a linear sequence out of this quadratic one by subtracting the $2n^2$ term from each term. So for example, the first term will be $5-2n^2=5-2(1)^2=3.$ the second term will be: $11-2n^2=11-2(2)^2=3.$ And so on for each term of the sequence. As you can see, the result is a linear sequence represented by the constant 3 (it never changes).

Adding the results together, the formula will be $2n^2+3.$ You can check this formula by substituting with n for each term and comparing with the sequence you have. So if n=4, the term will be $2n^2+3=2(4)^2+3=2(16)+3=35$ which is correct.

Now for the missing term, it is the 6th term. So substitute in the formula by n=6: term= $2(6)^2+3=75$

Answer 2

The next number is 75.

The formula use here is and increase in the sequence by 4.

5 - 11 = 6

22 - 11 = 10

35 - 22 = 14

53 - 35 = 18

You can see the gradual increase in the sequence by 4.

The next number will be 53 +(18+4).

Which is 53 + 22

Which equals 75.

Hope this helps.