Question

Carmen is playing with blocks. She arranges stacks of blocks so that each successive level of blocks has 1 fewer block than the level below it, and the top level has 1 block. Such a stack with 3 levels is shown below. Carmen wants to make such a stack with 12 levels. How many blocks would she use to build this stack?

A. 66

B. 78

C. 132

D. 144

Answer 1

Carmen would require a total of 78 blocks to construct a stack comprising 12 levels, each following the pattern of diminishing by one block per level (option B). This calculation is derived from the sum of an arithmetic sequence, considering the blocks arranged in a decreasing order from 1 to 12.

Step-by-step explanation:

The number of blocks Carmen would use to build a stack with 12 levels follows a pattern where each level has 1 fewer block than the level below it. This creates an arithmetic sequence: 1, 2, 3, 4, ..., where the
`\(n^(th)\)` term of this sequence is n (the number of blocks in each level).

The formula for the sum of the first n terms of an arithmetic sequence is
`\(S_n = (n)/(2)(a_1 + a_n)\)`, where
`\(S_n\)`is the sum, n is the number of terms,
`\(a_1\)` is the first term, and
`\(a_n\)` is the
`\(n^(th)\)`term.

For Carmen's stack with 12 levels, the first term
`\(a_1\)` is 1 block and the
`\(12^(th)\)` term
`\(a_(12)\)` is 12 blocks.

Substituting these values into the formula:

`\[ S_(12)` = 12/2(1 + 12) = 6 × 13 = 78

Therefore, Carmen would use 78 blocks to build a stack with 12 levels.