‘n Kwadraatiese ry het die volgende eienskappe: Die tweede verskil is 10: Die eerste twee terme is gelyk, d.w.s. $T_1 = T_2$. $T_1 + T_2 + T_3 = 28$ 3.1 Toon dat die algemene term van die ry $T_n = 5n^2 - 15n + 16$ is. 3.2 Is 216 ‘n term van hierdie ry? Motiveer jou antwoord met die nodige berekening.

NSC Mathematics Number patterns, sequences and series: ‘n Kwadraatiese ry het die volge

‘n Kwadraatiese ry het die volgende eienskappe: Die tweede verskil is 10: Die eerste twee terme is gelyk, d.w.s - NSC Mathematics - Question 3 - 2023 - Paper 1

Question 3

‘n Kwadraatiese ry het die volgende eienskappe: Die tweede verskil is 10: Die eerste twee terme is gelyk, d.w.s. $T_1 = T_2$. $T_1 + T_2 + T_3 = 28$ 3.1 Toon dat... show full transcript

Worked Solution & Example Answer:‘n Kwadraatiese ry het die volgende eienskappe: Die tweede verskil is 10: Die eerste twee terme is gelyk, d.w.s - NSC Mathematics - Question 3 - 2023 - Paper 1

Step 1

3.1 Toon dat die algemene term van die ry $T_n = 5n^2 - 15n + 16$ is.

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Answer

To solve for the general term of the quadratic sequence, we start with the information given:

The second difference is 10, which indicates that the coefficient of $n^2$ is half of the second difference. Thus, we have: $2a = 10 \Rightarrow a = 5$
The first two terms are equal, meaning: $T_1 = T_2$
The sum of the first three terms: $T_1 + T_2 + T_3 = 28$

Using the general form of a quadratic sequence:
$T_n = an^2 + bn + c$ Substituting $a = 5$ : $T_n = 5n^2 + bn + c$

We know:

$T_1 = 5(1^2) + b(1) + c = 5 + b + c$
$T_2 = 5(2^2) + b(2) + c = 20 + 2b + c$

Since $T_1 = T_2$ , we can set the equations equal: $5 + b + c = 20 + 2b + c$ This simplifies to: $b = 15$

Plugging $b$ back into $T_n$ gives us: $T_n = 5n^2 - 15n + c$ Now substituting into the equation for the sum: $T_1 + T_2 + T_3 = 5 + 15 + c = 28$ Sufficient to determine: $20 + c = 28 \Rightarrow c = 8$
Therefore, the general term is: $T_n = 5n^2 - 15n + 16$

Step 2

3.2 Is 216 ‘n term van hierdie ry? Motiveer jou antwoord met die nodige berekening.

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Answer

To determine if 216 is a term in the sequence given by: $T_n = 5n^2 - 15n + 16$ we set: $T_n = 216$

This leads to the equation: $5n^2 - 15n + 16 = 216$ Subtracting 216 from both sides gives: $5n^2 - 15n - 200 = 0$

To simplify, we divide the entire equation by 5: $n^2 - 3n - 40 = 0$

Factoring the quadratic equation: $(n - 8)(n + 5) = 0$ Thus: $n = 8 \quad \text{or} \quad n = -5$

Since $n$ must be a positive integer, we conclude: $n = 8$ Hence, 216 is indeed a term of the quadratic sequence corresponding to $n = 8$ .

‘n Kwadraatiese ry het die volgende eienskappe: Die tweede verskil is 10: Die eerste twee terme is gelyk, d.w.s - NSC Mathematics - Question 3 - 2023 - Paper 1

Worked Solution & Example Answer:‘n Kwadraatiese ry het die volgende eienskappe: Die tweede verskil is 10: Die eerste twee terme is gelyk, d.w.s - NSC Mathematics - Question 3 - 2023 - Paper 1

3.1 Toon dat die algemene term van die ry $T_n = 5n^2 - 15n + 16$ is.

3.2 Is 216 ‘n term van hierdie ry? Motiveer jou antwoord met die nodige berekening.

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