a staircase contains a total of 120 blocks and the top step contains 1 block. How many steps would have been using these blocks of every step upward is decreased by 2 block from the previous step?

Question

Please show step by step please

Answer 1

The number of stones in the steps are the odd numbers: 1, 3, 5, .....

The sum of the n odd numbers is 2n - 1

120 = 2n - 1

121 = 2n

Obviously an error in the data.

If, as Stanton D. suggests, the top step (one stone) is not counted:

121 = 2n - 1

122 = 2n

61 = n

There are 61 steps .

Answer 2

Hi Arthur S.,

So what you must do here first, is to set up a sketch. The staircase has 1 block for the top step, 3 blocks for the next-to-top step, 5 blocks for the third step down, and so on. Your sketch should have all the steps aligned on the right edge, and be stepped outwards towards the left edge 2 blocks for each step downwards, from the previous step above. Now what geometric figure does this downwards-growing staircase represent? It's a right triangle, of course. The trick is how to transform this sketch into an algebraic expression ("desired algebraic expression") for the total number of blocks in the staircase, given as an input (variable) the height of the staircase.

There is a general method for doing this task, called the method of successive differences. To do it, start by writing, in two columns next to each other, the height of the staircase (in column 1) and the total number of blocks in the growing staircase, (in column 2). Your columns should look like the following, with column headers:

Staircase Total steps First Difference Second Difference

height in staircase

1 1

3

2 4 2

5

3 9 2

7

4 16 2

9

5 25

Note that I have added two additional columns, representing the "first difference" = Total blocks(n+1) -Total blocks(n), and "second difference" = First difference(n+1) - First difference(n) .

Inspect the table to see; the (n) or (n+1) represent the indexed position, or subscript, of the titled variable, that is, how far down you are in that particular column.

As you go progressively across from left-to-right in the table, the various columns represent more and more broad comparisons of the total blocks data up-and-down in the table.

Note that the second difference is a constant value, all the way up and down in the table!

Now, here is the mathematically important information: each of the successive difference columns (left-to-right) in the table corresponds to a successively higher power of the initial variable (height of staircase). When you achieve a constant value in a particular rank of difference, that indicates that you have discovered a term in your algebraic expression answer. In particular here, you have found out that your algebraic expression contains a term "an^2", where a is a constant, and n is your staircase height. To discover what value a has here, you do this process over again, with n^2 as the values in your data column (which is column 2 in the example above). Let's see what that gives us:

Data (n^2) First diff. Second Diff

1

3

4 2

5

9 2

7

16 2

9

25

You see that the value for the second difference = 2 matches the one in the original example. Since it does, that shows that the n^2 is used exactly once in the algebraic expression we are building, that is, a = 1 .

All right, we now have one term of the desired algebraic expression. So we accept that term, and subtract it (as a series!) out from the total blocks series we started with. Now what that look like is:

Total blocks n^2 "Residual

series term series term series term"

1 1 0

4 4 0

9 9 0

OK, we can see that there is nothing left to account for -- the 1n^2 term did the trick for the entire "desired algebraic expression". (I included this step, because there is sometimes a residual series which must be tallied up by expressions of the form of bn^1 + c , in order to generate the proper original series).

Now we are ready to solve the original stated problem. If the staircase was as described, in a perfect triangular shape, the number of blocks would have to satisfy the expression n^2, for some value of n. But we immediately discover, to our shock, that 120 is not such a value, but that 121 is (=11^2). So, the staircase is 11 steps high, and the initial problem statement was sloppy, it should have stated that the total number of blocks in the staircase was 120, NOT INCLUDING THE TOP STEP.

So, Arthur S., you've gotten a hefty dose of "how to decompose a polynomial expression series". It's a rather powerful tool, even though it's simple to operate. If you want to explore further, try "decomposing" a series consisting of the blocks in a staircase that has TWO blocks as its top step, four as the next step down, and so forth. I think you will be in for a surprise!

-- Cheers, -- Mr. d.