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