Math Problem Statement
Solution
It appears you have uploaded two images showing a mathematical problem involving a pattern of yellow squares and corresponding questions. Let me analyze them and assist you with detailed solutions.
From the images:
- The pattern shows an incremental addition of squares.
- The questions involve:
- Determining the number of blocks for 10 figures.
- Predicting the number of blocks for the next figure.
- Counting the blocks for the 10th figure.
Let me solve step by step:
Step 1: Identify the pattern
- The sequence of figures shows an incremental increase in the number of blocks.
- Figure 1 has 1 block.
- Figure 2 has 3 blocks.
- Figure 3 has 6 blocks.
- Figure 4 has 10 blocks.
This forms a triangular number sequence:
Step 2: Total blocks needed for 10 figures
The sum of blocks across 10 figures can be calculated as:
= \sum_{n=1}^{10} \frac{n(n+1)}{2}$$ Using direct summation: $$\text{Sum} = 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 = 220$$ So, **220 blocks are needed to draw 10 figures.** --- ### **Step 3: Blocks in the next figure (5th figure)** Using the formula for triangular numbers: $$T_5 = \frac{5(5+1)}{2} = \frac{5 \cdot 6}{2} = 15$$ So, the next figure has **15 blocks.** --- ### **Step 4: Blocks in the 10th figure** Using the same formula: $$T_{10} = \frac{10(10+1)}{2} = \frac{10 \cdot 11}{2} = 55$$ So, the 10th figure contains **55 blocks.** --- ### Final Answers: 1. **220 blocks are needed to draw 10 figures.** 2. **The next figure (5th) has 15 blocks.** 3. **The 10th figure contains 55 blocks.** Would you like a detailed visualization of this pattern or further clarifications? --- **Relative Questions:** 1. How is the triangular number sequence derived? 2. How can this formula be applied to find the $$n$$-th term of any similar sequence? 3. What real-world examples apply triangular numbers? 4. How do summations like this extend to three-dimensional problems? 5. How does this relate to other number patterns, like square or Fibonacci numbers? **Tip:** Memorizing the triangular number formula can save time in solving sequential or summation problems!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Number Patterns
Sequences
Triangular Numbers
Formulas
T_n = n(n+1)/2
Theorems
Triangular Number Theorem
Suitable Grade Level
Grades 6-8
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