Sequence Word Problem

[0313phd]

Problem: The number of bricks in the bottom row of a brick wall is 49. The next row up from the bottom contains 47 bricks, and each subsequent row contains 2 fewer bricks than the row immediately below it. The number of bricks in the top row is 3. If the wall is one brick thick, what is the total number of bricks in the wall?

I am not getting the correct answer. Is this an arithmetic sequence? Sn= n/2 (a1 +an)
so that since there are 46 steps between the bottom and top steps, 46/2(3 + 49). However, the correct answer is 624. Is this a geometric sequence? Any help will be greatly appreciated. 0313

 

[soroban]

Hello,0313phd!

The number of bricks in the bottom row of a brick wall is 49.
The next row up from the bottom contains 47 bricks,
and each subsequent row contains 2 fewer bricks than the row immediately below it.
The number of bricks in the top row is 3.

If the wall is one brick thick, what is the total number of bricks in the wall?


I am not getting the correct answer.
Is this an arithmetic sequence? . Yes!
. . . \(\displaystyle S_n\:=\: \tfrac{n}{2}\left(a_1 + a_n\right)\)

So that since there are 46 steps between the bottom and top steps. . No, there are 24 rows.

Counting from top, we have:

. . \(\displaystyle \begin{array}{|c||c|c|c|c|c|c|} \hline \text{Row} & 1 & 2 & 3 & 4 & \hdots & 24 \\ \hline \text{Bricks} & 3 & 5 & 7 & 9 & \hdots & 49 \\ \hline\end{array}\)

\(\displaystyle \text{So we have: }\:S_{24} \;=\;\tfrac{24}{2}(3 + 49) \:=\12)(52) \:=\:624\)

 

[TchrWill]

Problem: The number of bricks in the bottom row of a brick wall is 49. The next row up from the bottom contains 47 bricks, and each subsequent row contains 2 fewer bricks than the row immediately below it. The number of bricks in the top row is 3. If the wall is one brick thick, what is the total number of bricks in the wall?

I am not getting the correct answer. Is this an arithmetic sequence? Sn= n/2 (a1 +an)
so that since there are 46 steps between the bottom and top steps, 46/2(3 + 49). However, the correct answer is 624. Is this a geometric sequence? Any help will be greatly appreciated. 0313

An alternate thought:

Add a brick atop those below and you have rows of 49, 47, 45, 43,...........7, 5, 3, 1.

The sum of the first "n" odd numbers is n^2.

Therefore, the sum of the 25 rows of bricks is 25^2 or 625.

Remove the single brick added for conveniance, and the total number of bricks is 624.

 

[JeffM]

Problem: The number of bricks in the bottom row of a brick wall is 49. The next row up from the bottom contains 47 bricks, and each subsequent row contains 2 fewer bricks than the row immediately below it. The number of bricks in the top row is 3. If the wall is one brick thick, what is the total number of bricks in the wall?

I am not getting the correct answer. Is this an arithmetic sequence? Sn= n/2 (a1 +an)
so that since there are 46 steps between the bottom and top steps, 46/2(3 + 49). However, the correct answer is 624. Is this a geometric sequence? Any help will be greatly appreciated. 0313

1 + 3 + 5 + ... + 49 = S[sub:241ucnbg]49[/sub:241ucnbg]
3 + 5 + ... + 49 = T[sub:241ucnbg]49[/sub:241ucnbg].
So, T[sub:241ucnbg]n[/sub:241ucnbg] = S[sub:241ucnbg]n[/sub:241ucnbg] - 1.

All clear so far?

There is a formula for the sum of the first n odd numbers, but maybe you do not know it. So derive it. (It is tricky to decide what to memorize. My tendency is to remember concepts and techniques and as few formulae as possible, but that is me. I was (sort of) trained as an historian so you cannot rely on me for math advice UNLESS I PROVE IT.)

Let's do a few examples. (This is not the only way to solve such problems, but, WHEN IT WORKS, it is the simplest.)

1 = 1.
1 + 3 = 4.
1 + 3 + 5 = 9.
1 + 3 + 5 + 7 = 16.

Do you see a pattern?

Not looking for a fancy proof of the pattern, but let's confirm it through an informal mathematical induction.

What is the standard formula for an odd number?

Assume [(2 * 1) - 1] + ... (2k - 1) = k[sup:241ucnbg]2[/sup:241ucnbg].
Then [(2 * 1) - 1] + ... [(2k) - 1] + [2(k + 1) - 1] = k[sup:241ucnbg]2[/sup:241ucnbg] + 2k + 2 - 1 = k[sup:241ucnbg]2[/sup:241ucnbg] + 2k + 1 = (k + 1)[sup:241ucnbg]2[/sup:241ucnbg].

Now can you solve your problem?

 

[0313phd]

THANKS, 0313

 

[Denis]

3 + 49 = 52
5 + 47 = 52
....