Mai V.

asked • 06/25/15# The figures below show solid figures formed with 1-cm cubes. (a) How many 1-cm cubes are needed to build Figure 5? (b) Which figure is made up of 204 cubes?

## 2 Answers By Expert Tutors

Figure N has N^2 + (N-1)^2 + … + 1 cubes

So, figure 5 has 5^2 + 4^2 + 3^2 + 2^2 + 1^2 = 25 + 16 + 9 + 4 + 1 = 55 cubes

Now, the sum of the squares of the numbers from 1 to n is

N^2 + (N-1)^2 + (N-2)^2 + … + 1^2 =

**N*(N+1)*(2*N+1)/6**( proof not given here)

So, to find the N for which there are 204 cubes:

204 = N*(N+1)*(2*N+1)/6

(again, not solving a cubic equation),

When

**N=8**8*9*17 / 6 = 204

**guess**an N that, when cubed, then doubled, then divided by 6 equals 9999. Let's

**guess**14, because 2*14^3/6 ≈ 915. Let's

**check**: 14*15*29/6 = 1015. The 14th term is 1015.

p.s., you might want to watch: https://www.youtube.com/watch?v=3IX-ukfgZdk

Mai V.

*thingy thingy*thing:

**^**

06/25/15

David W.

^{2.. }

06/25/15

David W.

**guess**an N that, when cubed, then doubled, then divides by 6 equals the number. Let's guess 14, because 2*14^3/6 = 915. Let's check: 14*15*29/6 = 1015.

**The 14th term is 1015.**

06/25/15

Mai V.

**Primary School**

*(or "Elementary School" in US)*, and this method seems like

*Secondary School*

*(or "High School" in US)*. Any other methods easier for me?

06/26/15

David W.

06/26/15

Son N. answered • 06/25/15

Math and Physics Tutor

^{2}+ 3

^{2}+ 4

^{2}+ 5

^{2}+ 6

^{2}cubes. This equals 91

^{2 }cubes, so we simply need to add 49 to our previous sum, 91.

^{2 }= 140 + 64 = 204

Mai V.

*Guess & Check*.

*9999*cubes, am I going to do one by one?

**No, definitely I won't**, and I believe there'll be a fixed solution solution.

06/25/15

David W.

**N*(N+1)*(2*N+1)/6**(not explained here), you could

**guess**a number that when cubed and the result doubled, then divided by 6 approximates 9999 (the equation resembles 2*N

^{3}/6). Ah, that is 14, because (2*14^3) / 6 = 915 (this is the guess part). And,

**checking**, 14*15*29/6 = 1015. The 14th term is 1015.

06/25/15

David W.

**guess**an N that when cubed, doubled, then divided by 6 (the formula is approximately 2*N ^3/6) produces 9999. Let's see, 2*14^3/6 = 915, so let's

**check**:

**The 14th term is 1015.**

06/25/15

David W.

**guess**an N that, when cubed, then doubled, then divides by 6 equals 9999. Let's

**guess**14, because 2*14^3/6 = 915. Let's

**check**: 14*15*29/6 = 1015. The 14th term is 1015.

p.s., you might want to watch: https://www.youtube.com/watch?v=3IX-ukfgZdk

06/25/15

Son N.

^{th}term:

_{n}= 1 + 2

^{2}+ 3

^{2}+ ... + n

^{2}

_{n}= n

^{3}/3 + n

^{2}/2 + n/6

_{n}= 204 you should that n = 8 works perfectly. However, finding the exact solution to n is a little difficult as this is a third degree polynomial. Since this equation clearly can't be factor, finding the exact solution for n is beyond my level of expertise.

**how**to guess.

06/26/15

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Mark M.

06/25/15