Guess the rule and add the next number in the sequence.

1, 5, 13, 25, 41,?

Suspect: we start adding 4, later 8, 12, 16 ...

Therefore, the next number is 41 + 20 = 61.

Now, we try to guess the rule.

Note that, the numbers that we add are multiples of 4. Try to write the sequence in a different way like this:

1, 1 + 4, (1+4) + 8, ((1+4) + 8) + 12, (((1+4) + 8) + 12) + 16, ...

1, 1+4, 1 + 4 (1+2), 1 + 4 (1+2+3), 1 + 4 (1+2+3+4) ...

You must to know that 1+2+3 + ... + n = n (n+1) / 2

Then, the nth numbers of the sequence must to be 1 + 4n (n+1) / 2 = 1+2n (n+1) = a_n

Check the formula:

n = 0 then a_0 = 1 + 0

n=1 then a_1 = 1 + 2 (2) = 5

n=2 then a_2 = 1 + 2 (2) (3) = 13

n=3 then a_3 = 1 + 2 (3) (4) = 25

n=4 then a_4 = 1 + 2 (4) (5) = 41

n=5 then a_5 = 1 + 2 (5) (6) = 61

...