Chapter 7, Problem 7TP

Solution

To identify: The sum that describes the number of blocks shown.

  

The options are:

  $\left(a\right){\displaystyle \sum _{i=2}^{20}\left(i+1\right)}$

  $\left(b\right){\displaystyle \sum _{i=1}^4\left(i+1\right)}$

  $\left(c\right){\displaystyle \sum _{i=2}^{20}i\left(i+1\right)}$

  $\left(d\right){\displaystyle \sum _{i=1}^{4}i\left(i+1\right)}$

The correct option is $\left(d\right){\displaystyle \sum _{i=1}^{4}i\left(i+1\right)}$ .

Given information:

The number of blocks in the nth layer, where $n=1$

represents the top layer. The given diagram is:

  

Explanation:

Consider the given diagram,

  

It can be seen that there are 4 layers in the diagram. The pattern followed by the sack is $2,6,12,20$ .

The rule that describe the blocks in the nth layer is $a_n=n\left(n+1\right)$ .

Then the sum of the series is given by $2+6+12+20$ .

In summation notation, the index of summation $i$ varies from 1 to 4.

It can be written as: ${\displaystyle \sum _{i=1}^{4}i\left(i+1\right)}$

Therefore, the sum is ${\displaystyle \sum _{i=1}^{4}i\left(i+1\right)}$ . Option $\left(d\right){\displaystyle \sum _{i=1}^{4}i\left(i+1\right)}$ should be the correct answer.