Fibonacci numbers Take 10 numbers as shown below:
a,b (a+b), (a+2b), (2a+3b), (3a+5b), (5a+8b), (8a+13b), (13a+21b), and (21a+34b). Sum of all these numbers =11(5a+8b) `=11 xx 7"th number"`. Taking a =8, b=13, write 10 Fibonacci numbers and verify that sum of all these numbers `=11 xx 7"th number".`

Answer

To solve the problem, we will follow the steps outlined in the question and verify the sum of the Fibonacci numbers generated using the given values of \( a \) and \( b \).

### Step-by-Step Solution:

1. **Identify the values of \( a \) and \( b \)**:
- Given \( a = 8 \) and \( b = 13 \).

2. **Calculate the Fibonacci numbers using the formulas**:
- The Fibonacci numbers are calculated as follows:
- First number: \( a = 8 \)
- Second number: \( b = 13 \)
- Third number: \( a + b = 8 + 13 = 21 \)
- Fourth number: \( a + 2b = 8 + 2 \times 13 = 8 + 26 = 34 \)
- Fifth number: \( 2a + 3b = 2 \times 8 + 3 \times 13 = 16 + 39 = 55 \)
- Sixth number: \( 3a + 5b = 3 \times 8 + 5 \times 13 = 24 + 65 = 89 \)
- Seventh number: \( 5a + 8b = 5 \times 8 + 8 \times 13 = 40 + 104 = 144 \)
- Eighth number: \( 8a + 13b = 8 \times 8 + 13 \times 13 = 64 + 169 = 233 \)
- Ninth number: \( 13a + 21b = 13 \times 8 + 21 \times 13 = 104 + 273 = 377 \)
- Tenth number: \( 21a + 34b = 21 \times 8 + 34 \times 13 = 168 + 442 = 610 \)

3. **List the Fibonacci numbers**:
- The 10 Fibonacci numbers are:
- 1st: \( 8 \)
- 2nd: \( 13 \)
- 3rd: \( 21 \)
- 4th: \( 34 \)
- 5th: \( 55 \)
- 6th: \( 89 \)
- 7th: \( 144 \)
- 8th: \( 233 \)
- 9th: \( 377 \)
- 10th: \( 610 \)

4. **Calculate the sum of these Fibonacci numbers**:
- Sum = \( 8 + 13 + 21 + 34 + 55 + 89 + 144 + 233 + 377 + 610 \)
- Sum = \( 8 + 13 = 21 \)
- Sum = \( 21 + 21 = 42 \)
- Sum = \( 42 + 34 = 76 \)
- Sum = \( 76 + 55 = 131 \)
- Sum = \( 131 + 89 = 220 \)
- Sum = \( 220 + 144 = 364 \)
- Sum = \( 364 + 233 = 597 \)
- Sum = \( 597 + 377 = 974 \)
- Sum = \( 974 + 610 = 1584 \)

5. **Verify the sum against \( 11 \times \text{7th number} \)**:
- The 7th number is \( 144 \).
- Calculate \( 11 \times 144 = 1584 \).

6. **Conclusion**:
- The sum of all the Fibonacci numbers is \( 1584 \), which equals \( 11 \times 144 \).