Name: Ilia

Who is asking: Student
Level: All

Question:
What the formula and explonation for formula for next patterns:

1) 1, 4, 10, 20, 35, 56, 84,...

2) 0, 6, 30, 90, 210, 420,...

Thanks!

Hi Ilia

One way to approach this type of problem is to take differences of successive terms to form a new sequence, then do the same for those until we come to a sequence of equal differences or to a pattern you recognize and can continue.

For example if the given sequence is 0, 2, 8, 21, 44, 80 then this procedure yields a constant row of three 3's. Extending this row to a fourth 3 allows the previous rows to be extended to give 132 at the next term in the original sequence.

       0,  2,  8,   21,   44,   80,  132
        +2  +6   +13   +23   +36  +52
          +4   +7   +10   +13  +16
             +3   +3    +3  +3

The same technique will work for the second sequence. Here the first row of differences is a constant times some binomial coefficients.

Cheers
Penny