The following sequence is a combination of an arithmetic sequence and a geometric sequence: $$ 5 ; 5 ; 15 ; 10 ; 25 ; 20 ; 35 ; 40 ; \ldots $$ $$ 4.1 \quad $$ Write down the next 2 terms. 4.2 Calculate $$ \mathrm{T}_{42}-\mathrm{T}_{41} $$. 4.3 Determine the sum of the first 50 terms. 4.4 Prove that ALL the terms of this infinite sequence will be divisible by 5 . (4)

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Let's analyze the given sequence:

$$
5 ;\ 5 ;\ 15 ;\ 10 ;\ 25 ;\ 20 ;\ 35 ;\ 40 ;\ \ldots
$$

Upon inspection, the sequence alternates between two distinct patterns:

1. **Odd-Indexed Terms (Arithmetic Sequence):**
   - **Terms:** $$ T_1 = 5 $$, $$ T_3 = 15 $$, $$ T_5 = 25 $$, $$ T_7 = 35 $$, $$\ldots$$
   - **Pattern:** Each odd-indexed term increases by 10.
   - **General Formula:** $$ T_n = 5n $$ for odd $$ n $$.

2. **Even-Indexed Terms (Geometric Sequence):**
   - **Terms:** $$ T_2 = 5 $$, $$ T_4 = 10 $$, $$ T_6 = 20 $$, $$ T_8 = 40 $$, $$\ldots$$
   - **Pattern:** Each even-indexed term doubles the previous even term.
   - **General Formula:** $$ T_n = 5 \times 2^{(n/2 - 1)} $$ for even $$ n $$.

### 4.1 Write Down the Next 2 Terms

Following the established patterns:

- **Next Odd-Indexed Term ($$ T_9 $$):**
  $$
  T_9 = 5 \times 9 = 45
  $$

- **Next Even-Indexed Term ($$ T_{10} $$):**
  $$
  T_{10} = 5 \times 2^{(10/2 - 1)} = 5 \times 2^{4} = 5 \times 16 = 80
  $$

**Answer:** The next two terms are **45** and **80**.
The next two terms are 45 and 80. ### 4.2 Calculate $$ \mathrm{T}_{42} - \mathrm{T}_{41} $$ - **$$ \mathrm{T}_{41} $$:** Since 41 is odd, it follows the arithmetic sequence pattern. $$ \mathrm{T}_{41} = 5 \times 41 = 205 $$ - **$$ \mathrm{T}_{42} $$:** Since 42 is even, it follows the geometric sequence pattern. $$ \mathrm{T}_{42} = 5 \times 2^{(42/2 - 1)} = 5 \times 2^{20} = 5 \times 1,048,576 = 5,242,880 $$ **Calculation:** $$ \mathrm{T}_{42} - \mathrm{T}_{41} = 5,242,880 - 205 = 5,242,675 $$ **Answer:** $$ \mathrm{T}_{42} - \mathrm{T}_{41} = 5,242,675 $$. ### 4.3 Determine the Sum of the First 50 Terms To find the sum of the first 50 terms, we'll separate the sequence into its odd and even parts. - **Sum of Odd-Indexed Terms ($$ S_{\text{odd}} $$):** - There are 25 odd terms. - The sequence is an arithmetic series with the first term $$ a = 5 $$ and common difference $$ d = 10 $$. - **Sum Formula:** $$ S_{\text{odd}} = \frac{n}{2} \times (2a + (n-1)d) = \frac{25}{2} \times (2 \times 5 + 24 \times 10) = 12.5 \times (10 + 240) = 12.5 \times 250 = 3,125 $$ - **Sum of Even-Indexed Terms ($$ S_{\text{even}} $$):** - There are 25 even terms. - The sequence is a geometric series with the first term $$ a = 5 $$ and common ratio $$ r = 2 $$. - **Sum Formula:** $$ S_{\text{even}} = a \times \frac{r^n - 1}{r - 1} = 5 \times \frac{2^{25} - 1}{2 - 1} = 5 \times (33,554,432 - 1) = 5 \times 33,554,431 = 167,772,155 $$ **Total Sum:** $$ S_{\text{total}} = S_{\text{odd}} + S_{\text{even}} = 3,125 + 167,772,155 = 167,775,280 $$ **Answer:** The sum of the first 50 terms is **167,775,280**. ### 4.4 Prove That ALL the Terms of This Infinite Sequence Will Be Divisible by 5 - **Odd-Indexed Terms:** - Each term is given by $$ T_n = 5n $$. - Since $$ 5n $$ is always divisible by 5, all odd-indexed terms are divisible by 5. - **Even-Indexed Terms:** - Each term is given by $$ T_n = 5 \times 2^{(n/2 - 1)} $$. - $$ 5 \times 2^{(n/2 - 1)} $$ is always divisible by 5 because it includes the factor 5. **Conclusion:** Every term in the sequence is divisible by 5. **Answer:** All terms of the sequence are divisible by 5.

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