Math  /  Algebra

QuestionThe start of a sequence of patterns made from tiles is shown below. The numbers of tiles in the patterns form a quadratic sequence.
Write an expression for the number of tiles in the nth n^{\text {th }} pattern.

Studdy Solution

STEP 1

What is this asking? Find a formula that tells us how many tiles are in any pattern number. Watch out! There are two types of tiles, don't forget to count both!

STEP 2

1. Find the orange squares rule
2. Find the green triangles rule
3. Combine the rules

STEP 3

Alright, let's **crack this tile puzzle**!
Notice how the orange squares form a bigger and bigger square in each pattern?
In pattern 1, we have 11=11 \cdot 1 = 1 orange square.
In pattern 2, we have 22=42 \cdot 2 = 4 orange squares.
In pattern 3, it's 33=93 \cdot 3 = 9 orange squares, and in pattern 4, we've got 44=164 \cdot 4 = 16 orange squares.
See the pattern?

STEP 4

So, for pattern number nn, the number of orange squares is nn=n2n \cdot n = n^2.
Boom! **Nailed it**!

STEP 5

Now, let's look at those **groovy green triangles**.
In pattern 1, there's 11 triangle.
Pattern 2 has 22 triangles.
Pattern 3 rocks 33 triangles, and pattern 4 has 44.
This one is super straightforward!

STEP 6

For pattern number nn, we simply have nn green triangles. **Easy peasy**!

STEP 7

Okay, time to **bring it all together**!
We know the number of orange squares is n2n^2 and the number of green triangles is nn.
The total number of tiles in pattern nn is just the sum of these two.

STEP 8

So, our **final formula** for the number of tiles in pattern nn is n2+nn^2 + n. **Mic drop**!

STEP 9

The expression for the number of tiles in the nthn^{\text{th}} pattern is n2+nn^2 + n.

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