Question

A set of blocks contains blocks of heights 1, 2, and 4 centimeters. imagine constructing towers by piling blocks of different heights directly on top of one another. (a tower of height 6 cm could be obtained using six 1-cm blocks, three 2-cm blocks, one 2-cm block with one 4-cm block on top, one 4-cm block with one 2-cm block on top, and so forth.) let t(n) be the number of ways to construct a tower of height n cm using blocks from the set. (assume an unlimited supply of blocks of each size.)

a. Find t(₁), t(₂), t(₃) and t(₄)

Answer 1

To find t(₁), t(₂), t(₃), and t(₄), we need to determine the number of ways to construct a tower of each height using blocks from the given set. t(₁) = 1, t(₂) = 2, t(₃) = 4, and t(₄) = 7.

Step-by-step explanation:

In this problem, we are asked to find the number of ways to construct a tower of a certain height using blocks of different heights. Let's consider the given set of blocks: 1 cm, 2 cm, and 4 cm. To find t(₁), we need to figure out how many ways we can build a tower of height 1 cm. Since the smallest block is 1 cm, there is only one way to construct a tower of height 1 cm, which is by using a single 1 cm block. Therefore, t(₁) = 1.

Similarly, to find t(₂), we need to determine the number of ways to build a tower of height 2 cm. There are two possibilities: either we use two 1 cm blocks or a single 2 cm block. Therefore, t(₂) = 2.

Continuing in the same way, we find that t(₃) = 4 and t(₄) = 7.