2.1 Given the quadratic sequence: $$ 0 ; 5 ; 14 ; \ldots ; 779 ; 860 $$ 2.1.1 Write down the value of the $$ 4^{\mathrm{\omega}} $$ term, $$ T_{4} $$, of this sequence. 2.1.2 Determine an expression for the $$ n^{\omega \mathrm{m}} $$ term of this sequence. 2.1.3 Calculate the number of terms in the sequence. 2.2 Determine the sum of the whole numbers between 100 and 1000 which are divisible by 11 (1) (5) [13]

Question

UpStudy AI · Accepted Answer

**2.1 Analysis of the Quadratic Sequence**

We are given the sequence:
$$
0; \quad 5; \quad 14; \quad \dots; \quad 779; \quad 860
$$
Since it is quadratic, we assume the general term is of the form:
$$
T_n = an^2 + bn + c.
$$

1. **Determine $$ a $$, $$ b $$, and $$ c $$:**

Using the first three terms:
   
   - For $$ n = 1 $$: 
     $$
     a(1)^2 + b(1) + c = a + b + c = 0.
     $$
   
   - For $$ n = 2 $$:
     $$
     a(2)^2 + b(2) + c = 4a + 2b + c = 5.
     $$
   
   - For $$ n = 3 $$:
     $$
     a(3)^2 + b(3) + c = 9a + 3b + c = 14.
     $$

2. **Subtract to eliminate $$ c $$:**

- Subtract the equation for $$ n = 1 $$ from that for $$ n = 2 $$:
     $$
     (4a + 2b + c) - (a + b + c) = 5 - 0 \quad \Longrightarrow \quad 3a + b = 5. \quad (1)
     $$
   
   - Subtract the equation for $$ n = 2 $$ from that for $$ n = 3 $$:
     $$
     (9a + 3b + c) - (4a + 2b + c) = 14 - 5 \quad \Longrightarrow \quad 5a + b = 9. \quad (2)
     $$

3. **Solve for $$ a $$ and $$ b $$:**

Subtract equation (1) from (2):
   $$
   (5a + b) - (3a + b) = 9 - 5 \quad \Longrightarrow \quad 2a = 4,
   $$
   so
   $$
   a = 2.
   $$
   
   Substitute $$ a = 2 $$ into equation (1):
   $$
   3(2) + b = 5 \quad \Longrightarrow \quad 6 + b = 5,
   $$
   hence,
   $$
   b = -1.
   $$

4. **Determine $$ c $$:**

Substitute $$ a = 2 $$ and $$ b = -1 $$ into the equation for $$ n = 1 $$:
   $$
   2 - 1 + c = 0 \quad \Longrightarrow \quad 1 + c = 0,
   $$
   so
   $$
   c = -1.
   $$

Thus, the $$ n^{\text{th}} $$ term of the sequence is:
$$
T_n = 2n^2 - n - 1.
$$

---

**2.1.1 Finding the $$ 4^{\text{th}} $$ Term**

Substitute $$ n = 4 $$ into the general formula:
$$
T_4 = 2(4)^2 - 4 - 1 = 2(16) - 4 - 1 = 32 - 4 - 1 = 27.
$$

---

**2.1.2 Expression for the $$ n^{\text{th}} $$ Term**

The expression for the $$ n^{\text{th}} $$ term is:
$$
T_n = 2n^2 - n - 1.
$$

---

**2.1.3 Calculating the Number of Terms in the Sequence**

The last term given is 860. Set $$ T_n = 860 $$:
$$
2n^2 - n - 1 = 860.
$$
Rearrange the equation:
$$
2n^2 - n - 861 = 0.
$$

Use the quadratic formula:
$$
n = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 2 \cdot (-861)}}{2 \cdot 2} = \frac{1 \pm \sqrt{1 + 6888}}{4} = \frac{1 \pm \sqrt{6889}}{4}.
$$
Since $$ 83^2 = 6889 $$, we have:
$$
n = \frac{1 \pm 83}{4}.
$$
Taking the positive solution:
$$
n = \frac{1+83}{4} = \frac{84}{4} = 21.
$$

There are 21 terms in the sequence.

---

**2.2 Sum of Whole Numbers Between 100 and 1000 Divisible by 11**

1. **Determine the Smallest and Largest Multiples of 11 in the Range:**

- The smallest whole number in the range divisible by 11 is:
     $$
     11 \times 10 = 110 \quad (\text{since } 11 \times 9 = 99 < 100).
     $$
   - The largest whole number in the range divisible by 11 is:
     $$
     11 \times 90 = 990 \quad (\text{since } 11 \times 91 = 1001 > 1000).
     $$

2. **Determine the Number of Terms:**

The sequence of multiples of 11 is an arithmetic sequence with first term $$ a_1 = 110 $$, last term $$ a_n = 990 $$, and common difference $$ d = 11 $$.

The number of terms $$ n $$ is given by:
   $$
   n = \frac{990 - 110}{11} + 1 = \frac{880}{11} + 1 = 80 + 1 = 81.
   $$

3. **Calculate the Sum:**

Use the formula for the sum of an arithmetic series:
   $$
   S_n = \frac{n}{2} (a_1 + a_n).
   $$
   Substitute the values:
   $$
   S_{81} = \frac{81}{2} (110 + 990) = \frac{81}{2} \times 1100 = 81 \times 550 = 44550.
   $$

---

$$
\textbf{Final Answers:}
$$
$$
\text{2.1.1 } T_4 = 27.
$$
$$
\text{2.1.2 } T_n = 2n^2 - n - 1.
$$
$$
\text{2.1.3 The sequence has 21 terms.}
$$
$$
\text{2.2 Sum} = 44\,550.
$$
**2.1 Analysis of the Quadratic Sequence** Given the sequence: 0, 5, 14, ..., 779, 860. 1. **Determine the General Term:** - Assume the $$ n^{\text{th}} $$ term is $$ T_n = 2n^2 - n - 1 $$. 2. **Find the 4th Term:** - $$ T_4 = 2(4)^2 - 4 - 1 = 27 $$. 3. **Number of Terms:** - The last term is 860. - Using the general term: $$ 2n^2 - n - 1 = 860 $$. - Solving for $$ n $$ gives $$ n = 21 $$. 4. **Sum of Whole Numbers Between 100 and 1000 Divisible by 11:** - Smallest multiple: 110. - Largest multiple: 990. - Number of terms: 81. - Sum: 44,550. **Final Answers:** - 4th term: 27 - General term: $$ T_n = 2n^2 - n - 1 $$ - Number of terms: 21 - Sum: 44,550

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Algebra Questions