PLEASE HELP ASAP!!

Question

(Growing staircase problem of the month)

1. How many blocks are needed to make the fourth step?
2. How many blocks does it take to build just the twelfth step?
3. How many blocks in all are needed to make a staircase of ten steps?
4. A staircase has 105 blocks, how many stairs does it have?

( Sorry I don't have an image of the staircase, well obviously there is no add image option) but please someone help ASAP!!!!

If this can help for imagery, there are 6 blocks, one block step, 2 block steps, & three-block steps.

Answer 1

Sounds like a Fibonacci sequence.

a_n+1 = a_n + a_n-1

And the sum of Fibonacci numbers is a_n+2 - 1

1 2 3 5 8 13 21 34 55 89 144 233

1) For the fourth step, a₄ = 5

2) a₁₂ = 233

3) For the sum, n+2 = 12, so the sum is a₁₂-1 = 232 blocks.

4) 105 block are used to build more than one set of steps. So need to find combinations of Fibonacci numbers that add up to 105

89+24+2 works, so 3 stairs in the staircase. There may be others.