Questions: Each successive figure below is made of small triangles like the first one in the sequence. Conjecture the number of small triangles needed to make a. the 100th figure? b. the nth figure?

Each successive figure below is made of small triangles like the first one in the sequence. Conjecture the number of small triangles needed to make a. the 100th figure? b. the nth figure?

Transcript text: Each successive figure below is made of small triangles like the first one in the sequence. Conjecture the number of small triangles needed to make $\mathbf{a}$. the 100 th figure? b. the nth figure? ,

Solution

Solution Steps

Step 1: Find the pattern

The first figure has 1 triangle. The second figure has 4 triangles. The third figure has 9 triangles. Notice that these are perfect squares: 1 = 1², 4 = 2², and 9 = 3². The pattern appears to be that the _n_th figure has _n_² small triangles.

Step 2: Calculate the number of triangles in the 100th figure

Following the pattern, the 100th figure would have 100² = 10,000 small triangles.

Step 3: Express the number of triangles for the nth figure

The _n_th figure will have _n_² small triangles.