Given the quadratic sequence: 0; 17; 32; ... 3.1 Determine an expression for the general term, $T_n$, of the quadratic sequence. 3.2 Which terms in the quadratic sequence have a value of 56? 3.3 Hence, or otherwise, calculate the value of $$rac{ extstyle igg( extstyle igg( extstyle igg( extstyle igg( extstyle igg( extstyle igg( extstyle igg( extstyle igg( extstyle igg( extstyle igg( extstyle igg( extstyle igg( extstyle igg( extstyle igg( extstyle igg( extstyle igg( extstyle igg( extstyle igg( extstyle igg( m=0 igg)^{10}_{n=5} igg)}^{10}_{m=1}igg)}}^{10}}^{10}}^{10}}^{10}}^{10}}^{15}igg)}^{15} igg)T_n}$$ [1]

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Given the quadratic sequence: 0; 17; 32; .. - NSC Mathematics - Question 3 - 2017 - Paper 1

Question 3

Given the quadratic sequence: 0; 17; 32; ... 3.1 Determine an expression for the general term, $T_n$, of the quadratic sequence. 3.2 Which terms in the quadratic s... show full transcript

Worked Solution & Example Answer:Given the quadratic sequence: 0; 17; 32; .. - NSC Mathematics - Question 3 - 2017 - Paper 1

Step 1

Determine an expression for the general term, $T_n$, of the quadratic sequence.

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Answer

To find the general term of the quadratic sequence, we first compute the first and second differences:

From the given terms: 0, 17, 32,
- First differences: 17 - 0 = 17, 32 - 17 = 15 → Results: 17, 15
- Second differences: 15 - 17 = -2 → Constant second difference of -2.
Using the polynomial form for a quadratic sequence: $T_n = an^2 + bn + c$ The second difference is given by $2a = -2$ , thus $a = -1$ .
The first differences can be expressed as: $3a + b = 17$ Substituting $a = -1$ gives:

ightarrow b = 20$$

Now using the initial term, where $n=1$ ,

ightarrow c = -19$$

Hence, the general term is: $T_n = -n^2 + 20n - 19$

Step 2

Which terms in the quadratic sequence have a value of 56?

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Answer

To find the terms in the sequence that equal 56:

Set the general term equal to 56: $56 = -n^2 + 20n - 19$ Rearranging gives: $n^2 - 20n + 75 = 0$
Factoring the equation: $(n - 15)(n - 5) = 0$ Therefore, $n = 15$ or $n = 5$ .
Conclusion: The terms in the quadratic sequence that have a value of 56 are for $n = 5$ and $n = 15$ .

Step 3

Hence, or otherwise, calculate the value of $ \sum_{n=5}^{10} T_n - \sum_{n=1}^{15} T_n $.

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Answer

To calculate the summation,

First, identify the terms:
- Using symmetry, since $T_{10} = T_{15}$ and $T_{5} = T_{10}$ ,
- We find the terms:
- For $n=5: T_5 = 56$ ,
- For $n=10: T_{10} = 81$ ,
- For $n=15: T_{15} = 56$ .
Then, compute the summations: $\sum_{n=5}^{10} T_n = T_5 + T_6 + T_7 + T_8 + T_9 + T_{10}$ The respective terms will sum to:
- $T_5 = 56$ ,
- $T_6 = 72$ ,
- $T_7 = 80$ ,
- $T_8 = 81$ ,
- $T_9 = 80$ ,
- $T_{10} = 81$ . Total: $56 + 72 + 80 + 81 + 80 + 81 = 450$ .
Calculating (\sum_{n=1}^{15} T_n ): This will yield the total result of 81 using previous evaluations.
Finally, subtract the two sums: $450 - 81 = 369$

Given the quadratic sequence: 0; 17; 32; .. - NSC Mathematics - Question 3 - 2017 - Paper 1

Worked Solution & Example Answer:Given the quadratic sequence: 0; 17; 32; .. - NSC Mathematics - Question 3 - 2017 - Paper 1

Determine an expression for the general term, $T_n$, of the quadratic sequence.

Which terms in the quadratic sequence have a value of 56?

Hence, or otherwise, calculate the value of \( \sum_{n=5}^{10} T_n - \sum_{n=1}^{15} T_n \).

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