Extend the Pattern

Relevant wiki: Identifying Pattern Relationships

The pattern of the total number of blocks is $1,3,5,7...$ Inspecting the first four terms, we see that for the $n^{th}$ item, the total number of blocks ( $t$ ) can be determined as follows: $t=n*2-1$

1st term, n=1: $1*2-1=1$

2nd term, n=2: $2*2-1=3$

n=3: $3*2-1=5$

n=4: $4*2-1=7$

n=43: $43*2-1=85$

We just notice that the total of cubes in each figure represents the sum of the cubes of that figure and the preceding one, for instance, in figure 4, it is 4+3=7 cubes. Hence, in figure 43, the total number of cubes is 43+42= 85

As we can see, the sequence 1,3,5,7,... appears to be a arithmetic sequence. So we can use in finding an ${ n }^{ th } term$ by using the following equation: ${ a }_{ n }={ a }_{ 1 }+(n-1)d$ .

Whereas, ${ a }_{ n }$ is the 1st term which is 1.

And, n as the no. of terms which is 43.

And, d as the common difference of each preceding or succeeding term, which is 2 as 3-1=2.

Pluging in the values,

${ a }_{ n }=1+(43-1)2$

${ a }_{ n }=1+84$

${ a }_{ n }=85$

Therefore, the 43rd term is $\boxed{85}$ .

43*2 = 86 86-1 = 85

EASY

Suppose the cube pattern is 1-43 ,i.e 1,2,3,4,5...41,42,43 than the answer will be 43 . 1,2,3,4,5...41,42,43 ------( eq. I) converting the (equation I) into given pattern 1,3,5,7... 1 , 2 , 3 , 4 , 5 .......41, 42 , 43 (0 ,+1, +2,+3,+4....+40,+41,+42) =1,3,5,7,9...............81,83,85 (question pattern ) The answer will be 85.

The first term has a height of one cube, second term has two cubes, third has four cubes and so on. The length is equal to height minus one cube because it is used by the height. So, the 43rd term has 43+(43-1) or 85 cubes.