A quadratic sequence is defined with the following properties:

2.3 .1 Write down the value of:
(a)
(b)
2.3 .2 Calculate the value of if .

Ask by Warner Simpson. in South Africa

Jan 23,2025

Question

A quadratic sequence is defined with the following properties:

2.3 .1 Write down the value of:
(a)
(b)
2.3 .2 Calculate the value of if .

Ask by Warner Simpson. in South Africa

Jan 23,2025

UpStudy AI · Accepted Answer

Let's analyze the quadratic sequence step by step.

### 2.3.1 Differences in the Sequence

A **quadratic sequence** has the general form:
$$ T_n = an^2 + bn + c $$
The first differences ($$ d_n = T_n - T_{n-1} $$) form an **arithmetic sequence** since the second differences are constant.

Given:
$$
\begin{align*}
T_2 - T_1 &= 7 \
T_3 - T_2 &= 13 \
T_4 - T_3 &= 19 \
\end{align*}
$$

#### (a) $$ T_5 - T_4 $$

Observe that the first differences increase by 6 each time:
$$
\begin{align*}
d_2 &= 7 \
d_3 &= 13 \quad (7 + 6) \
d_4 &= 19 \quad (13 + 6) \
d_5 &= 25 \quad (19 + 6) \
\end{align*}
$$

**Answer:** $$ T_5 - T_4 = 25 $$

#### (b) $$ T_{70} - T_{69} $$

Using the arithmetic pattern:
$$
d_n = 6n - 5
$$
So,
$$
d_{70} = 6 \times 70 - 5 = 420 - 5 = 415
$$

**Answer:** $$ T_{70} - T_{69} = 415 $$

### 2.3.2 Calculating $$ T_{69} $$ Given $$ T_{89} = 23,594 $$

We know:
$$
d_n = T_n - T_{n-1} = 6n - 5
$$
To find a general formula for $$ T_n $$:
$$
T_n = T_1 + \sum_{k=2}^{n} d_k = T_1 + \sum_{k=2}^{n} (6k - 5)
$$
Calculating the sum:
$$
\begin{align*}
\sum_{k=2}^{n} (6k - 5) &= 6 \sum_{k=2}^{n} k - 5(n-1) \
&= 6 \left( \frac{n(n+1)}{2} - 1 \right) - 5(n-1) \
&= 3n(n+1) - 6 - 5n + 5 \
&= 3n^2 - 2n - 1 \
\end{align*}
$$
Thus, the general formula is:
$$
T_n = 3n^2 - 2n + (T_1 - 1)
$$

Given $$ T_{89} = 23,594 $$:
$$
\begin{align*}
3(89)^2 - 2(89) + (T_1 - 1) &= 23,594 \
3 \times 7,921 - 178 + T_1 - 1 &= 23,594 \
23,763 - 178 + T_1 - 1 &= 23,594 \
23,584 + T_1 &= 23,594 \
T_1 &= 10 \
\end{align*}
$$
Now, using $$ T_1 = 10 $$:
$$
T_n = 3n^2 - 2n + 9
$$
Calculate $$ T_{69} $$:
$$
\begin{align*}
T_{69} &= 3(69)^2 - 2(69) + 9 \
&= 3 \times 4,761 - 138 + 9 \
&= 14,283 - 138 + 9 \
&= 14,154 \
\end{align*}
$$

**Answer:** $$ T_{69} = 14,155 $$

---

**Summary:**
1. **2.3.1**
   - (a) $$ T_5 - T_4 = 25 $$
   - (b) $$ T_{70} - T_{69} = 415 $$
2. **2.3.2**
   - $$ T_{69} = 14,155 $$
**2.3.1** - (a) $$ T_5 - T_4 = 25 $$ - (b) $$ T_{70} - T_{69} = 415 $$ **2.3.2** - $$ T_{69} = 14,155 $$

A quadratic sequence is defined with the following properties:

2.3 .1 Write down the value of:
(a)
(b)
2.3 .2 Calculate the value of if .

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A quadratic sequence is defined with the following properties: 2.3 .1 Write down the value of: (a) (b) 2.3 .2 Calculate the value of if .

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A quadratic sequence is defined with the following properties:

2.3 .1 Write down the value of:
(a)
(b)
2.3 .2 Calculate the value of if .