Sequences and series + Geo

Look at these triangles with dots. How many dots are there in each? Compute the number of dots needed to make the next two triangles.

• 31 dots

• 36 dots

• 11 dots

• 26 dots

Write down the sequence(-s) of natural numbers leaving remainder 1 on division by 3.

• 4, 7, 10, ...

• 5, 8, 11,...

• 145, 148, 151, ...

• 2001, 2004, 2007, ...

Write down the first four terms of the sequence of natural numbers ending in 1 or 6.

You are making a house of cards similar to the one shown. Write a rule for the number of cards in the nth row if the top row is row 1.

• $a_n=3n$

• $a_n=a_1+n\left(d-1\right)$

• $a_n=3n-1$

• $a_n=3\left(n-1\right)$

Tell whether the sequence is arithmetic.

• 1, -2, -5, -8, -11, . . .

• 16, 14, 11, 6, 3, . . .

• 5, 14, 23, 32, 41, . . .

• $\frac{7}{4},\frac{5}{4},\frac{3}{4},-\frac{3}{4},-\frac{5}{4},...$

• 0.5, 1, 1.5, 2, 2.5, . . .

For a certain arithmetic sequence,  $d=4$   and  $a_{30}=57.$ What is a rule for the nth term of the sequence?

•  $a_n=-63-4n$  

•  $a_n=-59-4n$  

•  $a_n=-63+4n$  

•  $a_n=-59+4n$  

Sol LeWitt’s sculpture Four-Sided Pyramid in the National Gallery of Art Sculpture Garden is made of concrete blocks. As shown in the diagram, each layer has 8 more visible blocks than the layer in front of it.

When you view the pyramid from one corner, a total of 12 layers are visible. How many of the pyramid’s blocks are visible?

At a restaurant, rectangular tables are placed together along their shared edges, as shown in the diagram below. How many people can be seated around 8 tables arranged in this way?

• 30 people

• 32 people

• 34 people

• 36 people

Tell whether the sequence is geometric.

• 1, 4, 8, 16, 32, . . .

• 4, 16, 64, 256, 1024, . . .

• 216, 36, 6, 1, 1/6, . . .

• -4, 12, -36, 108, -324, . . .

A pendulum that is released to swing freely travels 18 cm on the first swing. On each successive swing, the pendulum travels 80% of the distance of the previous swing. What is the total distance the pendulum swings?