To solve this problem, let's start by identifying the 10 numbers in the sequence given the initial values [tex]\(a = 8\)[/tex] and [tex]\(b = 13\)[/tex].
The sequence is built in a Fibonacci-like pattern, where each term is the sum of the previous two terms. Let's generate the 10 numbers step-by-step:
1. The starting values are:
[tex]\[
\text{First number} = a = 8
\][/tex]
[tex]\[
\text{Second number} = b = 13
\][/tex]
2. The next terms are calculated as follows:
[tex]\[
\text{Third number} = a + b = 8 + 13 = 21
\][/tex]
[tex]\[
\text{Fourth number} = a + 2b = 8 + 2 \cdot 13 = 8 + 26 = 34
\][/tex]
[tex]\[
\text{Fifth number} = 2a + 3b = 2 \cdot 8 + 3 \cdot 13 = 16 + 39 = 55
\][/tex]
[tex]\[
\text{Sixth number} = 3a + 5b = 3 \cdot 8 + 5 \cdot 13 = 24 + 65 = 89
\][/tex]
[tex]\[
\text{Seventh number} = 5a + 8b = 5 \cdot 8 + 8 \cdot 13 = 40 + 104 = 144
\][/tex]
[tex]\[
\text{Eighth number} = 8a + 13b = 8 \cdot 8 + 13 \cdot 13 = 64 + 169 = 233
\][/tex]
[tex]\[
\text{Ninth number} = 13a + 21b = 13 \cdot 8 + 21 \cdot 13 = 104 + 273 = 377
\][/tex]
[tex]\[
\text{Tenth number} = 21a + 34b = 21 \cdot 8 + 34 \cdot 13 = 168 + 442 = 610
\][/tex]
So, the 10 numbers in the sequence are:
[tex]\[ 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 \][/tex]
Next, let's calculate the sum of these numbers:
[tex]\[
8 + 13 + 21 + 34 + 55 + 89 + 144 + 233 + 377 + 610 = 1584
\][/tex]
Now, according to the problem, we need to verify that the sum of these numbers is equal to [tex]\(11\)[/tex] times the 7th number in the sequence. The 7th number in our sequence is:
[tex]\[ 144 \][/tex]
Calculating:
[tex]\[
11 \times 144 = 1584
\][/tex]
So indeed:
[tex]\[
\text{Sum of all these numbers} = 11 \times \text{Seventh number}
\][/tex]
Thus, we have verified that the sum of the 10 numbers in the sequence is [tex]\(1584\)[/tex], which is equal to [tex]\(11\)[/tex] times the 7th number [tex]\(144\)[/tex].
