picture 1 has 2 red dots and one yellow dot; picture 2 has 4 red dots and two yellow dots; picture 3 has 7 red dots and 3 yellow dots; picture 4 has 11 red dots and 4 yellow dots; Lizo plans to continue the pattern.
2.5.4 Describe Picture 60 and Picture 70 in words. (4)
2.5.5 Calculate the number of dots in Picture 60 and Picture 70. Show your 
work. 

To describe Picture 60 and Picture 70 and calculate the number of dots in each, we need to identify the patterns in the sequences of red and yellow dots. 

Let's first analyze the given pattern:

- Picture 1: 2 red dots, 1 yellow dot
- Picture 2: 4 red dots, 2 yellow dots
- Picture 3: 7 red dots, 3 yellow dots
- Picture 4: 11 red dots, 4 yellow dots

### Step 1: Identifying the Pattern

**Red Dots:**
- Picture 1: 2 red dots
- Picture 2: 4 red dots
- Picture 3: 7 red dots
- Picture 4: 11 red dots

The sequence of red dots is 2, 4, 7, 11, ...

To find the pattern, let's look at the differences between consecutive terms:
- 4 - 2 = 2
- 7 - 4 = 3
- 11 - 7 = 4

The differences between the numbers are increasing by 1 each time:
- 2, 3, 4, ...

This suggests that the \(n\)-th term of the red dots sequence is given by:
\[ a_n = a_{n-1} + (n+1) \]
with \(a_1 = 2\).

**Yellow Dots:**
- Picture 1: 1 yellow dot
- Picture 2: 2 yellow dots
- Picture 3: 3 yellow dots
- Picture 4: 4 yellow dots

The sequence of yellow dots is simply:
\[ 1, 2, 3, 4, ... \]

This suggests that the \(n\)-th term of the yellow dots sequence is given by:
\[ b_n = n \]

### Step 2: Finding the General Formula

For red dots:
- The initial term \(a_1 = 2\).
- The sequence follows \(a_n = a_{n-1} + (n+1)\).

For yellow dots:
- The sequence follows \(b_n = n\).

### Step 3: Calculating Specific Terms

To find \(a_{60}\) and \(a_{70}\):

For the red dots, we need to sum the sequence incrementally up to the 60th and 70th terms. 

#### Calculating \(a_{60}\):

Using the recursive pattern:
\[ a_n = a_1 + \sum_{k=2}^{n} (k+1) \]

\[ a_{60} = 2 + (3 + 4 + 5 + ... + 61) \]

The sum of an arithmetic series is given by:
\[ \text{Sum} = \frac{n}{2} ( \text{first term} + \text{last term} ) \]
where \(n = 59\) terms (from 3 to 61).

\[ \text{Sum} = \frac{59}{2} \times (3 + 61) = \frac{59}{2} \times 64 = 59 \times 32 = 1888 \]

Thus,
\[ a_{60} = 2 + 1888 = 1890 \]

#### Calculating \(a_{70}\):

\[ a_{70} = 2 + (3 + 4 + 5 + ... + 71) \]

\[ \text{Sum} = \frac{69}{2} \times (3 + 71) = \frac{69}{2} \times 74 = 69 \times 37 = 2553 \]

Thus,
\[ a_{70} = 2 + 2553 = 2555 \]

For the yellow dots:
\[ b_{60} = 60 \]
\[ b_{70} = 70 \]

### Step 4: Describing the Pictures

**Picture 60:**
- Number of red dots: 1890
- Number of yellow dots: 60

**Picture 70:**
- Number of red dots: 2555
- Number of yellow dots: 70

### Step 5: Total Number of Dots

For Picture 60:
\[ \text{Total dots} = 1890 + 60 = 1950 \]

For Picture 70:
\[ \text{Total dots} = 2555 + 70 = 2625 \]

### Summary

- **Picture 60**: It has 1890 red dots and 60 yellow dots, making a total of 1950 dots.
- **Picture 70**: It has 2555 red dots and 70 yellow dots, making a total of 2625 dots.

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Explore how to describe and calculate the number of dots in Picture 60 and Picture 70 based on a given pattern. Understand the sequences of red and yellow dots and apply arithmetic series summation to find specific terms. Suitable for advanced high school students.

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